\(\int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx\) [379]

Optimal result

Integrand size = 24, antiderivative size = 312 \[ \int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx=\frac {2 \left (b c^2-a d^2\right ) e \sqrt {e x}}{b^2}+\frac {4 c d (e x)^{3/2}}{3 b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}+\frac {\sqrt [4]{a} \left (b c^2+2 \sqrt {a} \sqrt {b} c d-a d^2\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{9/4}}-\frac {\sqrt [4]{a} \left (b c^2+2 \sqrt {a} \sqrt {b} c d-a d^2\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{9/4}}-\frac {\sqrt [4]{a} \left (b c^2-2 \sqrt {a} \sqrt {b} c d-a d^2\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} b^{9/4}} \] Output:

2*(-a*d^2+b*c^2)*e*(e*x)^(1/2)/b^2+4/3*c*d*(e*x)^(3/2)/b+2/5*d^2*(e*x)^(5/ 
2)/b/e+1/2*a^(1/4)*(b*c^2+2*a^(1/2)*b^(1/2)*c*d-a*d^2)*e^(3/2)*arctan(1-2^ 
(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/b^(9/4)-1/2*a^(1/4)*(b* 
c^2+2*a^(1/2)*b^(1/2)*c*d-a*d^2)*e^(3/2)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1 
/2)/a^(1/4)/e^(1/2))*2^(1/2)/b^(9/4)-1/2*a^(1/4)*(b*c^2-2*a^(1/2)*b^(1/2)* 
c*d-a*d^2)*e^(3/2)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(e*x)^(1/2)/e^(1/2)/(a^ 
(1/2)+b^(1/2)*x))*2^(1/2)/b^(9/4)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.68 \[ \int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx=\frac {(e x)^{3/2} \left (4 \sqrt [4]{b} \sqrt {x} \left (15 b c^2-15 a d^2+10 b c d x+3 b d^2 x^2\right )-15 \sqrt {2} \sqrt [4]{a} \left (-b c^2-2 \sqrt {a} \sqrt {b} c d+a d^2\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+15 \sqrt {2} \sqrt [4]{a} \left (-b c^2+2 \sqrt {a} \sqrt {b} c d+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{30 b^{9/4} x^{3/2}} \] Input:

Integrate[((e*x)^(3/2)*(c + d*x)^2)/(a + b*x^2),x]
 

Output:

((e*x)^(3/2)*(4*b^(1/4)*Sqrt[x]*(15*b*c^2 - 15*a*d^2 + 10*b*c*d*x + 3*b*d^ 
2*x^2) - 15*Sqrt[2]*a^(1/4)*(-(b*c^2) - 2*Sqrt[a]*Sqrt[b]*c*d + a*d^2)*Arc 
Tan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 15*Sqrt[2]* 
a^(1/4)*(-(b*c^2) + 2*Sqrt[a]*Sqrt[b]*c*d + a*d^2)*ArcTanh[(Sqrt[2]*a^(1/4 
)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(30*b^(9/4)*x^(3/2))
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.26, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {559, 27, 552, 27, 552, 27, 554, 1482, 27, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx\)

\(\Big \downarrow \) 559

\(\displaystyle \frac {2 \int \frac {5 (e x)^{3/2} \left (b c^2+2 b d x c-a d^2\right )}{2 \left (b x^2+a\right )}dx}{5 b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(e x)^{3/2} \left (b c^2+2 b d x c-a d^2\right )}{b x^2+a}dx}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 552

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-\frac {2 e \int \frac {3 b \sqrt {e x} \left (2 a c d-\left (b c^2-a d^2\right ) x\right )}{2 \left (b x^2+a\right )}dx}{3 b}}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \int \frac {\sqrt {e x} \left (2 a c d-\left (b c^2-a d^2\right ) x\right )}{b x^2+a}dx}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 552

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (-\frac {2 e \int -\frac {a \left (b c^2+2 b d x c-a d^2\right )}{2 \sqrt {e x} \left (b x^2+a\right )}dx}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {a e \int \frac {b c^2+2 b d x c-a d^2}{\sqrt {e x} \left (b x^2+a\right )}dx}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 554

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \int \frac {\left (b c^2-a d^2\right ) e+2 b c d x e}{b x^2 e^2+a e^2}d\sqrt {e x}}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \left (\frac {\left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \int \frac {\sqrt {b} \left (\sqrt {a} e-\sqrt {b} e x\right )}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {a} \sqrt {b}}+\frac {\left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \int \frac {\sqrt {b} \left (\sqrt {b} x e+\sqrt {a} e\right )}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {a} \sqrt {b}}\right )}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \left (\frac {\left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {a}}+\frac {\left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {a}}\right )}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \left (\frac {\left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {a}}+\frac {\left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \left (\frac {\left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {a}}+\frac {\left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {a}}\right )}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \left (\frac {\left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {a}}+\frac {\left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {a}}\right )}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \left (\frac {\left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {a}}+\frac {\left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {a}}\right )}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \left (\frac {\left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {a}}+\frac {\left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {a}}\right )}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \left (\frac {\left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {e}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {e}}\right )}{2 \sqrt {a}}+\frac {\left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {a}}\right )}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {4}{3} c d (e x)^{3/2}-e \left (\frac {2 a e \left (\frac {\left (2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {a}}+\frac {\left (-2 \sqrt {a} \sqrt {b} c d-a d^2+b c^2\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {a}}\right )}{b}-\frac {2 \sqrt {e x} \left (b c^2-a d^2\right )}{b}\right )}{b}+\frac {2 d^2 (e x)^{5/2}}{5 b e}\)

Input:

Int[((e*x)^(3/2)*(c + d*x)^2)/(a + b*x^2),x]
 

Output:

(2*d^2*(e*x)^(5/2))/(5*b*e) + ((4*c*d*(e*x)^(3/2))/3 - e*((-2*(b*c^2 - a*d 
^2)*Sqrt[e*x])/b + (2*a*e*(((b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*(-(Arc 
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])]/(Sqrt[2]*a^(1/4)*b^ 
(1/4)*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e]) 
]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e])))/(2*Sqrt[a]) + ((b*c^2 - 2*Sqrt[a]*Sq 
rt[b]*c*d - a*d^2)*(-1/2*Log[Sqrt[a]*e + Sqrt[b]*e*x - Sqrt[2]*a^(1/4)*b^( 
1/4)*Sqrt[e]*Sqrt[e*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e]) + Log[Sqrt[a]*e 
+ Sqrt[b]*e*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e]*Sqrt[e*x]]/(2*Sqrt[2]*a^(1 
/4)*b^(1/4)*Sqrt[e])))/(2*Sqrt[a])))/b))/b
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 552

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[d*(e*x)^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[e 
/(b*(m + 2*p + 2))   Int[(e*x)^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 
2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && GtQ[m, 0] && NeQ[ 
m + 2*p + 2, 0] && (IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 554

Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] 
 :> Simp[2   Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] 
 /; FreeQ[{a, b, c, d, e}, x]
 

rule 559

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( 
m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1))   Int[(e*x)^m*(a + b* 
x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 
)*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && IGtQ[n, 1] &&  !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.11

method	result	size
risch	\(-\frac {2 \left (-3 b \,x^{2} d^{2}-10 b c d x +15 a \,d^{2}-15 b \,c^{2}\right ) x \,e^{2}}{15 b^{2} \sqrt {e x}}-\frac {a \left (\frac {\left (-a \,d^{2} e +b \,c^{2} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \,e^{2}}+\frac {d c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{2 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right ) e^{2}}{b^{2}}\)	\(346\)
derivativedivides	\(\frac {-\frac {2 \left (-\frac {b \,d^{2} \left (e x \right )^{\frac {5}{2}}}{5}-\frac {2 b c d e \left (e x \right )^{\frac {3}{2}}}{3}+a \,d^{2} e^{2} \sqrt {e x}-b \,c^{2} e^{2} \sqrt {e x}\right )}{b^{2}}+\frac {2 a \,e^{3} \left (\frac {\left (a \,d^{2} e -b \,c^{2} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}-\frac {d c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}}{e}\)	\(363\)
default	\(\frac {-\frac {2 \left (-\frac {b \,d^{2} \left (e x \right )^{\frac {5}{2}}}{5}-\frac {2 b c d e \left (e x \right )^{\frac {3}{2}}}{3}+a \,d^{2} e^{2} \sqrt {e x}-b \,c^{2} e^{2} \sqrt {e x}\right )}{b^{2}}+\frac {2 a \,e^{3} \left (\frac {\left (a \,d^{2} e -b \,c^{2} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}-\frac {d c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}}{e}\)	\(363\)
pseudoelliptic	\(-\frac {\left (-\frac {\left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}\, \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {\frac {a \,e^{2}}{b}}}{2}+4 \left (a \,d^{2}-\left (\frac {1}{5} d^{2} x^{2}+\frac {2}{3} c d x +c^{2}\right ) b \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}+\left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) d e \sqrt {2}\, a c \right ) e}{2 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} b^{2}}\)	\(380\)

Input:

int((e*x)^(3/2)*(d*x+c)^2/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-2/15*(-3*b*d^2*x^2-10*b*c*d*x+15*a*d^2-15*b*c^2)*x/b^2/(e*x)^(1/2)*e^2-a/ 
b^2*(1/4*(-a*d^2*e+b*c^2*e)*(a*e^2/b)^(1/4)/a/e^2*2^(1/2)*(ln((e*x+(a*e^2/ 
b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e*x-(a*e^2/b)^(1/4)*(e*x)^( 
1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2 
)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))+1/2*d*c/(a*e^2/b)^(1 
/4)*2^(1/2)*(ln((e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/ 
(e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2 
)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1 
/2)-1)))*e^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1618 vs. \(2 (233) = 466\).

Time = 0.16 (sec) , antiderivative size = 1618, normalized size of antiderivative = 5.19 \[ \int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(3/2)*(d*x+c)^2/(b*x^2+a),x, algorithm="fricas")
 

Output:

-1/30*(15*b^2*sqrt(-(b^4*sqrt(-(a*b^4*c^8 - 12*a^2*b^3*c^6*d^2 + 38*a^3*b^ 
2*c^4*d^4 - 12*a^4*b*c^2*d^6 + a^5*d^8)*e^6/b^9) + 4*(a*b*c^3*d - a^2*c*d^ 
3)*e^3)/b^4)*log((b^4*c^8 - 4*a*b^3*c^6*d^2 - 10*a^2*b^2*c^4*d^4 - 4*a^3*b 
*c^2*d^6 + a^4*d^8)*sqrt(e*x)*e^4 + (2*b^7*c*d*sqrt(-(a*b^4*c^8 - 12*a^2*b 
^3*c^6*d^2 + 38*a^3*b^2*c^4*d^4 - 12*a^4*b*c^2*d^6 + a^5*d^8)*e^6/b^9) + ( 
b^5*c^6 - 7*a*b^4*c^4*d^2 + 7*a^2*b^3*c^2*d^4 - a^3*b^2*d^6)*e^3)*sqrt(-(b 
^4*sqrt(-(a*b^4*c^8 - 12*a^2*b^3*c^6*d^2 + 38*a^3*b^2*c^4*d^4 - 12*a^4*b*c 
^2*d^6 + a^5*d^8)*e^6/b^9) + 4*(a*b*c^3*d - a^2*c*d^3)*e^3)/b^4)) - 15*b^2 
*sqrt(-(b^4*sqrt(-(a*b^4*c^8 - 12*a^2*b^3*c^6*d^2 + 38*a^3*b^2*c^4*d^4 - 1 
2*a^4*b*c^2*d^6 + a^5*d^8)*e^6/b^9) + 4*(a*b*c^3*d - a^2*c*d^3)*e^3)/b^4)* 
log((b^4*c^8 - 4*a*b^3*c^6*d^2 - 10*a^2*b^2*c^4*d^4 - 4*a^3*b*c^2*d^6 + a^ 
4*d^8)*sqrt(e*x)*e^4 - (2*b^7*c*d*sqrt(-(a*b^4*c^8 - 12*a^2*b^3*c^6*d^2 + 
38*a^3*b^2*c^4*d^4 - 12*a^4*b*c^2*d^6 + a^5*d^8)*e^6/b^9) + (b^5*c^6 - 7*a 
*b^4*c^4*d^2 + 7*a^2*b^3*c^2*d^4 - a^3*b^2*d^6)*e^3)*sqrt(-(b^4*sqrt(-(a*b 
^4*c^8 - 12*a^2*b^3*c^6*d^2 + 38*a^3*b^2*c^4*d^4 - 12*a^4*b*c^2*d^6 + a^5* 
d^8)*e^6/b^9) + 4*(a*b*c^3*d - a^2*c*d^3)*e^3)/b^4)) - 15*b^2*sqrt((b^4*sq 
rt(-(a*b^4*c^8 - 12*a^2*b^3*c^6*d^2 + 38*a^3*b^2*c^4*d^4 - 12*a^4*b*c^2*d^ 
6 + a^5*d^8)*e^6/b^9) - 4*(a*b*c^3*d - a^2*c*d^3)*e^3)/b^4)*log((b^4*c^8 - 
 4*a*b^3*c^6*d^2 - 10*a^2*b^2*c^4*d^4 - 4*a^3*b*c^2*d^6 + a^4*d^8)*sqrt(e* 
x)*e^4 + (2*b^7*c*d*sqrt(-(a*b^4*c^8 - 12*a^2*b^3*c^6*d^2 + 38*a^3*b^2*...
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 12.77 (sec) , antiderivative size = 925, normalized size of antiderivative = 2.96 \[ \int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**(3/2)*(d*x+c)**2/(b*x**2+a),x)
 

Output:

-9*a**(5/4)*d**2*e**(3/2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar( 
I*pi/4)/a**(1/4))*gamma(9/4)/(8*b**(9/4)*gamma(13/4)) + 9*I*a**(5/4)*d**2* 
e**(3/2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4 
))*gamma(9/4)/(8*b**(9/4)*gamma(13/4)) + 9*a**(5/4)*d**2*e**(3/2)*exp(-I*p 
i/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(9/4)/(8* 
b**(9/4)*gamma(13/4)) - 9*I*a**(5/4)*d**2*e**(3/2)*exp(-I*pi/4)*log(1 - b* 
*(1/4)*sqrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(9/4)/(8*b**(9/4)*gamma( 
13/4)) + 7*a**(3/4)*c*d*e**(3/2)*exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt(x)*e 
xp_polar(I*pi/4)/a**(1/4))*gamma(7/4)/(4*b**(7/4)*gamma(11/4)) + 7*I*a**(3 
/4)*c*d*e**(3/2)*exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/ 
4)/a**(1/4))*gamma(7/4)/(4*b**(7/4)*gamma(11/4)) - 7*a**(3/4)*c*d*e**(3/2) 
*exp(-3*I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gam 
ma(7/4)/(4*b**(7/4)*gamma(11/4)) - 7*I*a**(3/4)*c*d*e**(3/2)*exp(-3*I*pi/4 
)*log(1 - b**(1/4)*sqrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(7/4)/(4*b** 
(7/4)*gamma(11/4)) + 5*a**(1/4)*c**2*e**(3/2)*exp(-I*pi/4)*log(1 - b**(1/4 
)*sqrt(x)*exp_polar(I*pi/4)/a**(1/4))*gamma(5/4)/(8*b**(5/4)*gamma(9/4)) - 
 5*I*a**(1/4)*c**2*e**(3/2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_pola 
r(3*I*pi/4)/a**(1/4))*gamma(5/4)/(8*b**(5/4)*gamma(9/4)) - 5*a**(1/4)*c**2 
*e**(3/2)*exp(-I*pi/4)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/ 
4))*gamma(5/4)/(8*b**(5/4)*gamma(9/4)) + 5*I*a**(1/4)*c**2*e**(3/2)*exp...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x)^(3/2)*(d*x+c)^2/(b*x^2+a),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (233) = 466\).

Time = 0.14 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.52 \[ \int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx=-\frac {1}{60} \, e {\left (\frac {30 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e - \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e + 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4} e} + \frac {30 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e - \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e + 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4} e} + \frac {15 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e - \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e - 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{b^{4} e} - \frac {15 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e - \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e - 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{b^{4} e} - \frac {8 \, {\left (3 \, \sqrt {e x} b^{4} d^{2} e^{10} x^{2} + 10 \, \sqrt {e x} b^{4} c d e^{10} x + 15 \, \sqrt {e x} b^{4} c^{2} e^{10} - 15 \, \sqrt {e x} a b^{3} d^{2} e^{10}\right )}}{b^{5} e^{10}}\right )} \] Input:

integrate((e*x)^(3/2)*(d*x+c)^2/(b*x^2+a),x, algorithm="giac")
 

Output:

-1/60*e*(30*sqrt(2)*((a*b^3*e^2)^(1/4)*b^2*c^2*e - (a*b^3*e^2)^(1/4)*a*b*d 
^2*e + 2*(a*b^3*e^2)^(3/4)*c*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4 
) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(b^4*e) + 30*sqrt(2)*((a*b^3*e^2)^(1/4)* 
b^2*c^2*e - (a*b^3*e^2)^(1/4)*a*b*d^2*e + 2*(a*b^3*e^2)^(3/4)*c*d)*arctan( 
-1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(b^4 
*e) + 15*sqrt(2)*((a*b^3*e^2)^(1/4)*b^2*c^2*e - (a*b^3*e^2)^(1/4)*a*b*d^2* 
e - 2*(a*b^3*e^2)^(3/4)*c*d)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + 
 sqrt(a*e^2/b))/(b^4*e) - 15*sqrt(2)*((a*b^3*e^2)^(1/4)*b^2*c^2*e - (a*b^3 
*e^2)^(1/4)*a*b*d^2*e - 2*(a*b^3*e^2)^(3/4)*c*d)*log(e*x - sqrt(2)*(a*e^2/ 
b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(b^4*e) - 8*(3*sqrt(e*x)*b^4*d^2*e^10* 
x^2 + 10*sqrt(e*x)*b^4*c*d*e^10*x + 15*sqrt(e*x)*b^4*c^2*e^10 - 15*sqrt(e* 
x)*a*b^3*d^2*e^10)/(b^5*e^10))
 

Mupad [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 1785, normalized size of antiderivative = 5.72 \[ \int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx=\text {Too large to display} \] Input:

int(((e*x)^(3/2)*(c + d*x)^2)/(a + b*x^2),x)
 

Output:

(e*x)^(1/2)*((2*c^2*e)/b - (2*a*d^2*e)/b^2) - atan((a^4*d^4*e^6*(e*x)^(1/2 
)*((a^2*c*d^3*e^3)/b^4 - (a^2*d^4*e^3*(-a*b^9)^(1/2))/(4*b^9) - (a*c^3*d*e 
^3)/b^3 - (c^4*e^3*(-a*b^9)^(1/2))/(4*b^7) + (3*a*c^2*d^2*e^3*(-a*b^9)^(1/ 
2))/(2*b^8))^(1/2)*32i)/(32*a^3*c^5*d*e^8 - (192*a^4*c^3*d^3*e^8)/b + (16* 
a^2*c^6*e^8*(-a*b^9)^(1/2))/b^4 - (16*a^5*d^6*e^8*(-a*b^9)^(1/2))/b^7 + (3 
2*a^5*c*d^5*e^8)/b^2 - (112*a^3*c^4*d^2*e^8*(-a*b^9)^(1/2))/b^5 + (112*a^4 
*c^2*d^4*e^8*(-a*b^9)^(1/2))/b^6) + (a^2*b*c^4*e^6*(e*x)^(1/2)*((a^2*c*d^3 
*e^3)/b^4 - (a^2*d^4*e^3*(-a*b^9)^(1/2))/(4*b^9) - (a*c^3*d*e^3)/b^3 - (c^ 
4*e^3*(-a*b^9)^(1/2))/(4*b^7) + (3*a*c^2*d^2*e^3*(-a*b^9)^(1/2))/(2*b^8))^ 
(1/2)*32i)/((16*a^2*c^6*e^8*(-a*b^9)^(1/2))/b^5 - (192*a^4*c^3*d^3*e^8)/b^ 
2 - (16*a^5*d^6*e^8*(-a*b^9)^(1/2))/b^8 + (32*a^3*c^5*d*e^8)/b + (32*a^5*c 
*d^5*e^8)/b^3 - (112*a^3*c^4*d^2*e^8*(-a*b^9)^(1/2))/b^6 + (112*a^4*c^2*d^ 
4*e^8*(-a*b^9)^(1/2))/b^7) - (a^3*c^2*d^2*e^6*(e*x)^(1/2)*((a^2*c*d^3*e^3) 
/b^4 - (a^2*d^4*e^3*(-a*b^9)^(1/2))/(4*b^9) - (a*c^3*d*e^3)/b^3 - (c^4*e^3 
*(-a*b^9)^(1/2))/(4*b^7) + (3*a*c^2*d^2*e^3*(-a*b^9)^(1/2))/(2*b^8))^(1/2) 
*192i)/((16*a^2*c^6*e^8*(-a*b^9)^(1/2))/b^5 - (192*a^4*c^3*d^3*e^8)/b^2 - 
(16*a^5*d^6*e^8*(-a*b^9)^(1/2))/b^8 + (32*a^3*c^5*d*e^8)/b + (32*a^5*c*d^5 
*e^8)/b^3 - (112*a^3*c^4*d^2*e^8*(-a*b^9)^(1/2))/b^6 + (112*a^4*c^2*d^4*e^ 
8*(-a*b^9)^(1/2))/b^7))*(-(a^2*d^4*e^3*(-a*b^9)^(1/2) + b^2*c^4*e^3*(-a*b^ 
9)^(1/2) + 4*a*b^6*c^3*d*e^3 - 4*a^2*b^5*c*d^3*e^3 - 6*a*b*c^2*d^2*e^3*...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.56 \[ \int \frac {(e x)^{3/2} (c+d x)^2}{a+b x^2} \, dx =\text {Too large to display} \] Input:

int((e*x)^(3/2)*(d*x+c)^2/(b*x^2+a),x)
 

Output:

(sqrt(e)*e*(60*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 
 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c*d - 30*b**(3/4)*a**(1 
/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4) 
*a**(1/4)*sqrt(2)))*a*d**2 + 30*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a 
**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c**2 - 
 60*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)* 
sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c*d + 30*b**(3/4)*a**(1/4)*sqrt(2) 
*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*s 
qrt(2)))*a*d**2 - 30*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr 
t(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c**2 - 30*b**(1/4 
)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sq 
rt(b)*x)*b*c*d + 30*b**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4 
)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*c*d - 15*b**(3/4)*a**(1/4)*sqrt(2)*log( 
 - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*d**2 + 15*b* 
*(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) 
 + sqrt(b)*x)*b*c**2 + 15*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a 
**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*d**2 - 15*b**(3/4)*a**(1/4)*sqrt( 
2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*c**2 - 1 
20*sqrt(x)*a*b*d**2 + 120*sqrt(x)*b**2*c**2 + 80*sqrt(x)*b**2*c*d*x + 24*s 
qrt(x)*b**2*d**2*x**2))/(60*b**3)