\(\int \frac {(e x)^{7/2} (c+d x)}{(a+b x^2)^2} \, dx\) [402]

Optimal result

Integrand size = 22, antiderivative size = 297 \[ \int \frac {(e x)^{7/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {2 c e^3 \sqrt {e x}}{b^2}+\frac {2 d e^2 (e x)^{3/2}}{3 b^2}+\frac {a e^3 \sqrt {e x} (c+d x)}{2 b^2 \left (a+b x^2\right )}+\frac {\sqrt [4]{a} \left (5 \sqrt {b} c+7 \sqrt {a} d\right ) e^{7/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} b^{11/4}}-\frac {\sqrt [4]{a} \left (5 \sqrt {b} c+7 \sqrt {a} d\right ) e^{7/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} b^{11/4}}-\frac {\sqrt [4]{a} \left (5 \sqrt {b} c-7 \sqrt {a} d\right ) e^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} b^{11/4}} \] Output:

2*c*e^3*(e*x)^(1/2)/b^2+2/3*d*e^2*(e*x)^(3/2)/b^2+1/2*a*e^3*(e*x)^(1/2)*(d 
*x+c)/b^2/(b*x^2+a)+1/8*a^(1/4)*(5*b^(1/2)*c+7*a^(1/2)*d)*e^(7/2)*arctan(1 
-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/b^(11/4)-1/8*a^(1/4) 
*(5*b^(1/2)*c+7*a^(1/2)*d)*e^(7/2)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^ 
(1/4)/e^(1/2))*2^(1/2)/b^(11/4)-1/8*a^(1/4)*(5*b^(1/2)*c-7*a^(1/2)*d)*e^(7 
/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^(11/4)
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.67 \[ \int \frac {(e x)^{7/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {(e x)^{7/2} \left (\frac {4 b^{3/4} \sqrt {x} \left (4 b x^2 (3 c+d x)+a (15 c+7 d x)\right )}{a+b x^2}+3 \sqrt {2} \sqrt [4]{a} \left (5 \sqrt {b} c+7 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} \sqrt [4]{a} \left (-5 \sqrt {b} c+7 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{24 b^{11/4} x^{7/2}} \] Input:

Integrate[((e*x)^(7/2)*(c + d*x))/(a + b*x^2)^2,x]
 

Output:

((e*x)^(7/2)*((4*b^(3/4)*Sqrt[x]*(4*b*x^2*(3*c + d*x) + a*(15*c + 7*d*x))) 
/(a + b*x^2) + 3*Sqrt[2]*a^(1/4)*(5*Sqrt[b]*c + 7*Sqrt[a]*d)*ArcTan[(Sqrt[ 
a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 3*Sqrt[2]*a^(1/4)*(-5 
*Sqrt[b]*c + 7*Sqrt[a]*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[ 
a] + Sqrt[b]*x)]))/(24*b^(11/4)*x^(7/2))
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.27, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {549, 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.

\(\displaystyle \int \frac {(e x)^{7/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx\)

\(\Big \downarrow \) 549

\(\displaystyle \frac {e^2 \int \frac {(e x)^{3/2} (5 c+7 d x)}{2 \left (b x^2+a\right )}dx}{2 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \int \frac {(e x)^{3/2} (5 c+7 d x)}{b x^2+a}dx}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 552

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {2 e \int \frac {3 \sqrt {e x} (7 a d-5 b c x)}{2 \left (b x^2+a\right )}dx}{3 b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \int \frac {\sqrt {e x} (7 a d-5 b c x)}{b x^2+a}dx}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 552

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (-\frac {2 e \int -\frac {a b (5 c+7 d x)}{2 \sqrt {e x} \left (b x^2+a\right )}dx}{b}-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (a e \int \frac {5 c+7 d x}{\sqrt {e x} \left (b x^2+a\right )}dx-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 554

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \int \frac {5 c e+7 d x e}{b x^2 e^2+a e^2}d\sqrt {e x}-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}-7 d\right ) \int \frac {\sqrt {b} \left (\sqrt {a} e-\sqrt {b} e x\right )}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 b}+\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}+7 d\right ) \int \frac {\sqrt {b} \left (\sqrt {b} x e+\sqrt {a} e\right )}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 b}\right )-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}-7 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}+7 d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}\right )-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}-7 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}+7 d\right ) \left (\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}\right )}{2 \sqrt {b}}\right )-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}-7 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}+7 d\right ) \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}-7 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}+7 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}-7 d\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}+7 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}-7 d\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}+7 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}-7 d\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {e}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}+7 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {e^2 \left (\frac {14 d (e x)^{3/2}}{3 b}-\frac {e \left (2 a e \left (\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}+7 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {5 \sqrt {b} c}{\sqrt {a}}-7 d\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )-10 c \sqrt {e x}\right )}{b}\right )}{4 b}-\frac {e (e x)^{5/2} (c+d x)}{2 b \left (a+b x^2\right )}\)

Input:

Int[((e*x)^(7/2)*(c + d*x))/(a + b*x^2)^2,x]
 

Output:

-1/2*(e*(e*x)^(5/2)*(c + d*x))/(b*(a + b*x^2)) + (e^2*((14*d*(e*x)^(3/2))/ 
(3*b) - (e*(-10*c*Sqrt[e*x] + 2*a*e*((((5*Sqrt[b]*c)/Sqrt[a] + 7*d)*(-(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[b]) + (((5*Sqrt[b]*c)/Sqrt[a 
] - 7*d)*(-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[b]))))/b))/(4*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 549

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

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 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 = 350, normalized size of antiderivative = 1.18

method	result	size
risch	\(\frac {2 \left (d x +3 c \right ) x \,e^{4}}{3 b^{2} \sqrt {e x}}-\frac {a \left (\frac {-\frac {d \left (e x \right )^{\frac {3}{2}}}{2}-\frac {c e \sqrt {e x}}{2}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {5 c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 e a}+\frac {7 d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right ) e^{4}}{b^{2}}\)	\(350\)
derivativedivides	\(2 e^{2} \left (\frac {\frac {d \left (e x \right )^{\frac {3}{2}}}{3}+c e \sqrt {e x}}{b^{2}}-\frac {a \,e^{2} \left (\frac {-\frac {d \left (e x \right )^{\frac {3}{2}}}{4}-\frac {c e \sqrt {e x}}{4}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {5 c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 e a}+\frac {7 d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}\right )\)	\(354\)
default	\(2 e^{2} \left (\frac {\frac {d \left (e x \right )^{\frac {3}{2}}}{3}+c e \sqrt {e x}}{b^{2}}-\frac {a \,e^{2} \left (\frac {-\frac {d \left (e x \right )^{\frac {3}{2}}}{4}-\frac {c e \sqrt {e x}}{4}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {5 c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 e a}+\frac {7 d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}\right )\)	\(354\)
pseudoelliptic	\(-\frac {7 e^{2} \left (\frac {5 e b \sqrt {2}\, \left (b \,x^{2}+a \right ) c \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {\frac {a \,e^{2}}{b}}}{7}-\frac {8 d \left (e x \right )^{\frac {3}{2}} a b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{7}+\left (-\frac {40 \left (\frac {4 x^{2} \left (\frac {d x}{3}+c \right ) b}{5}+a \left (\frac {4 d x}{15}+c \right )\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} b \sqrt {e x}}{7}+d e \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {2}\, \left (b \,x^{2}+a \right ) a \right ) e \right )}{16 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} b^{3} \left (b \,x^{2}+a \right )}\)	\(370\)

Input:

int((e*x)^(7/2)*(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

2/3*(d*x+3*c)*x/b^2/(e*x)^(1/2)*e^4-a/b^2*(2*(-1/4*d*(e*x)^(3/2)-1/4*c*e*( 
e*x)^(1/2))/(b*e^2*x^2+a*e^2)+5/16*c/e*(a*e^2/b)^(1/4)/a*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))+7/16*d/b/(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*arcta 
n(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^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1011 vs. \(2 (211) = 422\).

Time = 0.18 (sec) , antiderivative size = 1011, normalized size of antiderivative = 3.40 \[ \int \frac {(e x)^{7/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((e*x)^(7/2)*(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

1/24*(3*(b^3*x^2 + a*b^2)*sqrt(-(70*a*c*d*e^7 + sqrt(-(625*a*b^2*c^4 - 245 
0*a^2*b*c^2*d^2 + 2401*a^3*d^4)*e^14/b^11)*b^5)/b^5)*log(-(625*b^2*c^4 - 2 
401*a^2*d^4)*sqrt(e*x)*e^10 + (7*sqrt(-(625*a*b^2*c^4 - 2450*a^2*b*c^2*d^2 
 + 2401*a^3*d^4)*e^14/b^11)*b^8*d + 5*(25*b^4*c^3 - 49*a*b^3*c*d^2)*e^7)*s 
qrt(-(70*a*c*d*e^7 + sqrt(-(625*a*b^2*c^4 - 2450*a^2*b*c^2*d^2 + 2401*a^3* 
d^4)*e^14/b^11)*b^5)/b^5)) - 3*(b^3*x^2 + a*b^2)*sqrt(-(70*a*c*d*e^7 + sqr 
t(-(625*a*b^2*c^4 - 2450*a^2*b*c^2*d^2 + 2401*a^3*d^4)*e^14/b^11)*b^5)/b^5 
)*log(-(625*b^2*c^4 - 2401*a^2*d^4)*sqrt(e*x)*e^10 - (7*sqrt(-(625*a*b^2*c 
^4 - 2450*a^2*b*c^2*d^2 + 2401*a^3*d^4)*e^14/b^11)*b^8*d + 5*(25*b^4*c^3 - 
 49*a*b^3*c*d^2)*e^7)*sqrt(-(70*a*c*d*e^7 + sqrt(-(625*a*b^2*c^4 - 2450*a^ 
2*b*c^2*d^2 + 2401*a^3*d^4)*e^14/b^11)*b^5)/b^5)) - 3*(b^3*x^2 + a*b^2)*sq 
rt(-(70*a*c*d*e^7 - sqrt(-(625*a*b^2*c^4 - 2450*a^2*b*c^2*d^2 + 2401*a^3*d 
^4)*e^14/b^11)*b^5)/b^5)*log(-(625*b^2*c^4 - 2401*a^2*d^4)*sqrt(e*x)*e^10 
+ (7*sqrt(-(625*a*b^2*c^4 - 2450*a^2*b*c^2*d^2 + 2401*a^3*d^4)*e^14/b^11)* 
b^8*d - 5*(25*b^4*c^3 - 49*a*b^3*c*d^2)*e^7)*sqrt(-(70*a*c*d*e^7 - sqrt(-( 
625*a*b^2*c^4 - 2450*a^2*b*c^2*d^2 + 2401*a^3*d^4)*e^14/b^11)*b^5)/b^5)) + 
 3*(b^3*x^2 + a*b^2)*sqrt(-(70*a*c*d*e^7 - sqrt(-(625*a*b^2*c^4 - 2450*a^2 
*b*c^2*d^2 + 2401*a^3*d^4)*e^14/b^11)*b^5)/b^5)*log(-(625*b^2*c^4 - 2401*a 
^2*d^4)*sqrt(e*x)*e^10 - (7*sqrt(-(625*a*b^2*c^4 - 2450*a^2*b*c^2*d^2 + 24 
01*a^3*d^4)*e^14/b^11)*b^8*d - 5*(25*b^4*c^3 - 49*a*b^3*c*d^2)*e^7)*sqr...
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 57.38 (sec) , antiderivative size = 2100, normalized size of antiderivative = 7.07 \[ \int \frac {(e x)^{7/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**(7/2)*(d*x+c)/(b*x**2+a)**2,x)
 

Output:

c*(45*a**(13/4)*b**3*e**(7/2)*x**(21/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar 
(I*pi/4)/a**(1/4))*gamma(9/4)/(32*a**3*b**(21/4)*x**(21/2)*exp(I*pi/4)*gam 
ma(13/4) + 32*a**2*b**(25/4)*x**(25/2)*exp(I*pi/4)*gamma(13/4)) - 45*I*a** 
(13/4)*b**3*e**(7/2)*x**(21/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4 
)/a**(1/4))*gamma(9/4)/(32*a**3*b**(21/4)*x**(21/2)*exp(I*pi/4)*gamma(13/4 
) + 32*a**2*b**(25/4)*x**(25/2)*exp(I*pi/4)*gamma(13/4)) - 45*a**(13/4)*b* 
*3*e**(7/2)*x**(21/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4 
))*gamma(9/4)/(32*a**3*b**(21/4)*x**(21/2)*exp(I*pi/4)*gamma(13/4) + 32*a* 
*2*b**(25/4)*x**(25/2)*exp(I*pi/4)*gamma(13/4)) + 45*I*a**(13/4)*b**3*e**( 
7/2)*x**(21/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamm 
a(9/4)/(32*a**3*b**(21/4)*x**(21/2)*exp(I*pi/4)*gamma(13/4) + 32*a**2*b**( 
25/4)*x**(25/2)*exp(I*pi/4)*gamma(13/4)) + 45*a**(9/4)*b**4*e**(7/2)*x**(2 
5/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/4)/a**(1/4))*gamma(9/4)/(32*a 
**3*b**(21/4)*x**(21/2)*exp(I*pi/4)*gamma(13/4) + 32*a**2*b**(25/4)*x**(25 
/2)*exp(I*pi/4)*gamma(13/4)) - 45*I*a**(9/4)*b**4*e**(7/2)*x**(25/2)*log(1 
 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(9/4)/(32*a**3*b**( 
21/4)*x**(21/2)*exp(I*pi/4)*gamma(13/4) + 32*a**2*b**(25/4)*x**(25/2)*exp( 
I*pi/4)*gamma(13/4)) - 45*a**(9/4)*b**4*e**(7/2)*x**(25/2)*log(1 - b**(1/4 
)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(9/4)/(32*a**3*b**(21/4)*x**( 
21/2)*exp(I*pi/4)*gamma(13/4) + 32*a**2*b**(25/4)*x**(25/2)*exp(I*pi/4)...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e x)^{7/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x)^(7/2)*(d*x+c)/(b*x^2+a)^2,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 [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.35 \[ \int \frac {(e x)^{7/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {e x} a d e^{5} x + \sqrt {e x} a c e^{5}}{2 \, {\left (b e^{2} x^{2} + a e^{2}\right )} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e^{3} + 7 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e^{3} + 7 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e^{3} - 7 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{16 \, b^{5}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e^{3} - 7 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{16 \, b^{5}} + \frac {2 \, {\left (\sqrt {e x} b^{4} d e^{3} x + 3 \, \sqrt {e x} b^{4} c e^{3}\right )}}{3 \, b^{6}} \] Input:

integrate((e*x)^(7/2)*(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

1/2*(sqrt(e*x)*a*d*e^5*x + sqrt(e*x)*a*c*e^5)/((b*e^2*x^2 + a*e^2)*b^2) - 
1/8*sqrt(2)*(5*(a*b^3*e^2)^(1/4)*b^2*c*e^3 + 7*(a*b^3*e^2)^(3/4)*d*e^2)*ar 
ctan(1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/ 
b^5 - 1/8*sqrt(2)*(5*(a*b^3*e^2)^(1/4)*b^2*c*e^3 + 7*(a*b^3*e^2)^(3/4)*d*e 
^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2/b)^ 
(1/4))/b^5 - 1/16*sqrt(2)*(5*(a*b^3*e^2)^(1/4)*b^2*c*e^3 - 7*(a*b^3*e^2)^( 
3/4)*d*e^2)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/b 
^5 + 1/16*sqrt(2)*(5*(a*b^3*e^2)^(1/4)*b^2*c*e^3 - 7*(a*b^3*e^2)^(3/4)*d*e 
^2)*log(e*x - sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/b^5 + 2/3 
*(sqrt(e*x)*b^4*d*e^3*x + 3*sqrt(e*x)*b^4*c*e^3)/b^6
 

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 780, normalized size of antiderivative = 2.63 \[ \int \frac {(e x)^{7/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(((e*x)^(7/2)*(c + d*x))/(a + b*x^2)^2,x)
 

Output:

atan((a^2*c^2*e^10*(e*x)^(1/2)*((49*a*d^2*e^7*(-a*b^11)^(1/2))/(64*b^11) - 
 (35*a*c*d*e^7)/(32*b^5) - (25*c^2*e^7*(-a*b^11)^(1/2))/(64*b^10))^(1/2)*5 
0i)/((343*a^4*d^3*e^14)/(4*b^4) - (125*a^2*c^3*e^14*(-a*b^11)^(1/2))/(4*b^ 
8) - (175*a^3*c^2*d*e^14)/(4*b^3) + (245*a^3*c*d^2*e^14*(-a*b^11)^(1/2))/( 
4*b^9)) - (a^3*d^2*e^10*(e*x)^(1/2)*((49*a*d^2*e^7*(-a*b^11)^(1/2))/(64*b^ 
11) - (35*a*c*d*e^7)/(32*b^5) - (25*c^2*e^7*(-a*b^11)^(1/2))/(64*b^10))^(1 
/2)*98i)/((343*a^4*d^3*e^14)/(4*b^3) - (125*a^2*c^3*e^14*(-a*b^11)^(1/2))/ 
(4*b^7) - (175*a^3*c^2*d*e^14)/(4*b^2) + (245*a^3*c*d^2*e^14*(-a*b^11)^(1/ 
2))/(4*b^8)))*(-(25*b*c^2*e^7*(-a*b^11)^(1/2) - 49*a*d^2*e^7*(-a*b^11)^(1/ 
2) + 70*a*b^6*c*d*e^7)/(64*b^11))^(1/2)*2i + atan((a^2*c^2*e^10*(e*x)^(1/2 
)*((25*c^2*e^7*(-a*b^11)^(1/2))/(64*b^10) - (35*a*c*d*e^7)/(32*b^5) - (49* 
a*d^2*e^7*(-a*b^11)^(1/2))/(64*b^11))^(1/2)*50i)/((343*a^4*d^3*e^14)/(4*b^ 
4) + (125*a^2*c^3*e^14*(-a*b^11)^(1/2))/(4*b^8) - (175*a^3*c^2*d*e^14)/(4* 
b^3) - (245*a^3*c*d^2*e^14*(-a*b^11)^(1/2))/(4*b^9)) - (a^3*d^2*e^10*(e*x) 
^(1/2)*((25*c^2*e^7*(-a*b^11)^(1/2))/(64*b^10) - (35*a*c*d*e^7)/(32*b^5) - 
 (49*a*d^2*e^7*(-a*b^11)^(1/2))/(64*b^11))^(1/2)*98i)/((343*a^4*d^3*e^14)/ 
(4*b^3) + (125*a^2*c^3*e^14*(-a*b^11)^(1/2))/(4*b^7) - (175*a^3*c^2*d*e^14 
)/(4*b^2) - (245*a^3*c*d^2*e^14*(-a*b^11)^(1/2))/(4*b^8)))*(-(49*a*d^2*e^7 
*(-a*b^11)^(1/2) - 25*b*c^2*e^7*(-a*b^11)^(1/2) + 70*a*b^6*c*d*e^7)/(64*b^ 
11))^(1/2)*2i + ((a*c*e^5*(e*x)^(1/2))/2 + (a*d*e^4*(e*x)^(3/2))/2)/(a*...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 637, normalized size of antiderivative = 2.14 \[ \int \frac {(e x)^{7/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((e*x)^(7/2)*(d*x+c)/(b*x^2+a)^2,x)
 

Output:

(sqrt(e)*e**3*(42*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)))*a*d + 42*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*d*x**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)))*a*c + 
 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*x**2 - 42*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)))*a*d - 42*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*d*x**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)))*a*c - 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*x**2 - 21*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqr 
t(2) + sqrt(a) + sqrt(b)*x)*a*d - 21*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*d*x**2 + 21*b**(1/4 
)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt( 
b)*x)*a*d + 21*b**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqr 
t(2) + sqrt(a) + sqrt(b)*x)*b*d*x**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*c + 15*b**(...