\(\int \frac {(c+d x)^2}{(e x)^{5/2} (a+b x^2)} \, dx\) [383]

Optimal result

Integrand size = 24, antiderivative size = 288 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^2}{3 a e (e x)^{3/2}}-\frac {4 c d}{a e^2 \sqrt {e x}}+\frac {\left (b c^2+2 \sqrt {a} \sqrt {b} c d-a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b} e^{5/2}}-\frac {\left (b c^2+2 \sqrt {a} \sqrt {b} c d-a d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b} e^{5/2}}-\frac {\left (b c^2-2 \sqrt {a} \sqrt {b} c d-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} a^{7/4} \sqrt [4]{b} e^{5/2}} \] Output:

-2/3*c^2/a/e/(e*x)^(3/2)-4*c*d/a/e^2/(e*x)^(1/2)+1/2*(b*c^2+2*a^(1/2)*b^(1 
/2)*c*d-a*d^2)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/ 
2)/a^(7/4)/b^(1/4)/e^(5/2)-1/2*(b*c^2+2*a^(1/2)*b^(1/2)*c*d-a*d^2)*arctan( 
1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(7/4)/b^(1/4)/e^( 
5/2)-1/2*(b*c^2-2*a^(1/2)*b^(1/2)*c*d-a*d^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)/a^(7/4)/b^(1/4)/e^(5/2 
)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.66 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\frac {x \left (-4 a^{3/4} \sqrt [4]{b} c (c+6 d x)+3 \sqrt {2} \left (b c^2+2 \sqrt {a} \sqrt {b} c d-a d^2\right ) x^{3/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-3 \sqrt {2} \left (b c^2-2 \sqrt {a} \sqrt {b} c d-a d^2\right ) x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{6 a^{7/4} \sqrt [4]{b} (e x)^{5/2}} \] Input:

Integrate[(c + d*x)^2/((e*x)^(5/2)*(a + b*x^2)),x]
 

Output:

(x*(-4*a^(3/4)*b^(1/4)*c*(c + 6*d*x) + 3*Sqrt[2]*(b*c^2 + 2*Sqrt[a]*Sqrt[b 
]*c*d - a*d^2)*x^(3/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/ 
4)*Sqrt[x])] - 3*Sqrt[2]*(b*c^2 - 2*Sqrt[a]*Sqrt[b]*c*d - a*d^2)*x^(3/2)*A 
rcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(6*a^(7/ 
4)*b^(1/4)*(e*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.39, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {559, 27, 553, 27, 553, 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.

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

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 553

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 553

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 554

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

Int[(c + d*x)^2/((e*x)^(5/2)*(a + b*x^2)),x]
 

Output:

(-2*d^2)/(3*b*e*(e*x)^(3/2)) + ((-2*(b*c^2 - a*d^2))/(3*a*e*(e*x)^(3/2)) + 
 (b*((-4*c*d)/(e*Sqrt[e*x]) - (2*(((b*c^2 + 2*Sqrt[a]*Sqrt[b]*c*d - a*d^2) 
*(-(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])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*S 
qrt[e])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e])))/(2*Sqrt[a]) + ((b*c^2 - 2*Sqr 
t[a]*Sqrt[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[Sqr 
t[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])))/e))/(a*e))/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 553

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp 
[1/(a*e*(m + 1))   Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 
2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
 

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 = 0.44 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {2 \left (6 d x +c \right ) c}{3 a x \sqrt {e x}\, e^{2}}-\frac {\frac {\left (-a \,d^{2} e +b \,c^{2} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \,e^{2}}+\frac {d c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{2 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{a \,e^{2}}\)	\(326\)
derivativedivides	\(\frac {-\frac {2 c^{2}}{3 a \left (e x \right )^{\frac {3}{2}}}-\frac {4 c d}{e a \sqrt {e x}}+\frac {2 \left (\frac {\left (a \,d^{2} e -b \,c^{2} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}-\frac {d c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a e}}{e}\)	\(335\)
default	\(\frac {-\frac {2 c^{2}}{3 a \left (e x \right )^{\frac {3}{2}}}-\frac {4 c d}{e a \sqrt {e x}}+\frac {2 \left (\frac {\left (a \,d^{2} e -b \,c^{2} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}-\frac {d c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a e}}{e}\)	\(335\)
pseudoelliptic	\(-\frac {-\frac {\left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}\, \left (e x \right )^{\frac {3}{2}} \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {\frac {a \,e^{2}}{b}}}{2}+e \left (d \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}\, \left (e x \right )^{\frac {3}{2}}+\frac {4 e \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \left (6 d x +c \right )}{3}\right ) a c}{2 \left (e x \right )^{\frac {3}{2}} \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} a^{2} e^{3}}\)	\(375\)

Input:

int((d*x+c)^2/(e*x)^(5/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-2/3*(6*d*x+c)*c/a/x/(e*x)^(1/2)/e^2-1/a*(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*arct 
an(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*ar 
ctan(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. 1604 vs. \(2 (213) = 426\).

Time = 0.17 (sec) , antiderivative size = 1604, normalized size of antiderivative = 5.57 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^2/(e*x)^(5/2)/(b*x^2+a),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(c+d x)^2}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (e x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )}\, dx \] Input:

integrate((d*x+c)**2/(e*x)**(5/2)/(b*x**2+a),x)
 

Output:

Integral((c + d*x)**2/((e*x)**(5/2)*(a + b*x**2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^2/(e*x)^(5/2)/(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. 434 vs. \(2 (213) = 426\).

Time = 0.14 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.51 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 \, {\left (6 \, c d e x + c^{2} e\right )}}{3 \, \sqrt {e x} a e^{3} x} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e - \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e + 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{2} e^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e - \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e + 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{2} e^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e - \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e - 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{4 \, a^{2} b^{2} e^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e - \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e - 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{4 \, a^{2} b^{2} e^{4}} \] Input:

integrate((d*x+c)^2/(e*x)^(5/2)/(b*x^2+a),x, algorithm="giac")
 

Output:

-2/3*(6*c*d*e*x + c^2*e)/(sqrt(e*x)*a*e^3*x) - 1/2*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)*arc 
tan(1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/( 
a^2*b^2*e^4) - 1/2*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))/(a^2*b^2*e^4) - 1/4*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))/(a^2* 
b^2*e^4) + 1/4*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))/(a^2*b^2*e^4)
 

Mupad [B] (verification not implemented)

Time = 8.41 (sec) , antiderivative size = 1706, normalized size of antiderivative = 5.92 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

int((c + d*x)^2/((e*x)^(5/2)*(a + b*x^2)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.76 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^2/(e*x)^(5/2)/(b*x^2+a),x)
 

Output:

(sqrt(e)*(12*sqrt(x)*b**(1/4)*a**(3/4)*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*d*x - 6*sqrt(x) 
*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqr 
t(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*d**2*x + 6*sqrt(x)*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*x - 12*sqrt(x)*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*x + 6*sqrt(x)*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*x - 6*sqrt(x)*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*x - 6*sqrt(x)*b**(1/4)*a**(3/4)*sq 
rt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*c* 
d*x + 6*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sq 
rt(2) + sqrt(a) + sqrt(b)*x)*b*c*d*x - 3*sqrt(x)*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*x 
+ 3*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqr 
t(2) + sqrt(a) + sqrt(b)*x)*b*c**2*x + 3*sqrt(x)*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*x - 3 
*sqrt(x)*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*x - 8*a*b*c**2 - 48*a*b*c*d*x))/(12*sqrt(x...