\(\int \frac {(e x)^{7/2}}{(c+d x)^{3/2} (a+b x^2)^2} \, dx\) [926]

Optimal result

Integrand size = 26, antiderivative size = 510 \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {c \left (b c^2-5 a d^2\right ) e (e x)^{5/2}}{2 a \left (b c^2+a d^2\right )^2 \sqrt {c+d x}}+\frac {\left (5 b c^2-a d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x}}{2 b \left (b c^2+a d^2\right )^2}-\frac {3 c d e^2 (e x)^{3/2} \sqrt {c+d x}}{\left (b c^2+a d^2\right )^2}+\frac {(e x)^{7/2} (a d+b c x)}{2 a \left (b c^2+a d^2\right ) \sqrt {c+d x} \left (a+b x^2\right )}-\frac {\left (5 \sqrt {-a} b^{3/2} c^3-8 a b c^2 d-\sqrt {-a} a \sqrt {b} c d^2-2 a^2 d^3\right ) e^{7/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 \sqrt [4]{-a} b^{3/2} \sqrt {\sqrt {b} c-\sqrt {-a} d} \left (b c^2+a d^2\right )^2}-\frac {\left (5 \sqrt {-a} b^{3/2} c^3+8 a b c^2 d+(-a)^{3/2} \sqrt {b} c d^2+2 a^2 d^3\right ) e^{7/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 \sqrt [4]{-a} b^{3/2} \sqrt {\sqrt {b} c+\sqrt {-a} d} \left (b c^2+a d^2\right )^2} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.07 (sec) , antiderivative size = 839, normalized size of antiderivative = 1.65 \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {e^3 \sqrt {e x} \left (b \sqrt {x} \left (-4 b^2 c^3 x^2+a^2 d^2 (c+d x)+a b c \left (-5 c^2+c d x+2 d^2 x^2\right )\right )-2 a d^{3/2} \sqrt {c+d x} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {2 b^2 c^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-15 a b c^2 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-16 a^2 d^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-12 b^2 c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-6 a b c d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+2 b^2 c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2+a b d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]+a d^{3/2} \sqrt {c+d x} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {31 a b c^2 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+32 a^2 d^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+6 b^2 c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+12 a b c d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-a b d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b c^3+3 b c^2 \text {$\#$1}+8 a d^2 \text {$\#$1}-3 b c \text {$\#$1}^2+b \text {$\#$1}^3}\&\right ]\right )}{2 b^2 \left (b c^2+a d^2\right )^2 \sqrt {x} \sqrt {c+d x} \left (a+b x^2\right )} \] Input:

Integrate[(e*x)^(7/2)/((c + d*x)^(3/2)*(a + b*x^2)^2),x]
 

Output:

-1/2*(e^3*Sqrt[e*x]*(b*Sqrt[x]*(-4*b^2*c^3*x^2 + a^2*d^2*(c + d*x) + a*b*c 
*(-5*c^2 + c*d*x + 2*d^2*x^2)) - 2*a*d^(3/2)*Sqrt[c + d*x]*(a + b*x^2)*Roo 
tSum[b*c^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*#1^2 - 4*b*c*#1^3 + b*#1 
^4 & , (2*b^2*c^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 
15*a*b*c^2*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 16* 
a^2*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 12*b^2*c^3 
*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - 6*a*b*c*d^2*Lo 
g[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 2*b^2*c^2*Log[c + 
 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2 + a*b*d^2*Log[c + 2*d* 
x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2)/(b*c^3 - 3*b*c^2*#1 - 8*a* 
d^2*#1 + 3*b*c*#1^2 - b*#1^3) & ] + a*d^(3/2)*Sqrt[c + d*x]*(a + b*x^2)*Ro 
otSum[b*c^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*#1^2 - 4*b*c*#1^3 + b*# 
1^4 & , (31*a*b*c^2*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - 
#1] + 32*a^2*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 6 
*b^2*c^3*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 12*a*b 
*c*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - a*b*d^2* 
Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2)/(-(b*c^3) + 3* 
b*c^2*#1 + 8*a*d^2*#1 - 3*b*c*#1^2 + b*#1^3) & ]))/(b^2*(b*c^2 + a*d^2)^2* 
Sqrt[x]*Sqrt[c + d*x]*(a + b*x^2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1831\) vs. \(2(510)=1020\).

Time = 6.49 (sec) , antiderivative size = 1831, normalized size of antiderivative = 3.59, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {615, 2009}

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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt [4]{-a} \left (\sqrt {b} c+\sqrt {-a} d\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right ) e^{7/2}}{2 b^{3/2} \sqrt {\sqrt {b} c-\sqrt {-a} d} \left (b c^2+a d^2\right )}-\frac {\sqrt [4]{-a} \left (7 \sqrt {b} c-4 \sqrt {-a} d\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right ) e^{7/2}}{4 b^{3/2} \left (\sqrt {b} c-\sqrt {-a} d\right )^{5/2}}+\frac {\left (3 b^{3/2} c^3-2 \sqrt {-a} b d c^2+5 a \sqrt {b} d^2 c+4 (-a)^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{7/2}}{4 a b^{3/2} d^{5/2} \left (\sqrt {b} c-\sqrt {-a} d\right )^2}+\frac {\left (3 b^{3/2} c^3+2 \sqrt {-a} b d c^2+5 a \sqrt {b} d^2 c+4 \sqrt {-a} a d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{7/2}}{4 a b^{3/2} d^{5/2} \left (\sqrt {b} c+\sqrt {-a} d\right )^2}+\frac {\left (\sqrt {b} c-\sqrt {-a} d\right ) \left (3 b c^2-4 \sqrt {-a} \sqrt {b} d c-8 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{7/2}}{16 a b^{3/2} d^{5/2} \left (b c^2+a d^2\right )}+\frac {\left (\sqrt {b} c+\sqrt {-a} d\right ) \left (3 b c^2+4 \sqrt {-a} \sqrt {b} d c-8 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{7/2}}{16 a b^{3/2} d^{5/2} \left (b c^2+a d^2\right )}-\frac {15 c^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right ) e^{7/2}}{8 a d^{5/2} \left (b c^2+a d^2\right )}-\frac {\sqrt [4]{-a} \left (7 \sqrt {b} c+4 \sqrt {-a} d\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right ) e^{7/2}}{4 b^{3/2} \left (\sqrt {b} c+\sqrt {-a} d\right )^{5/2}}+\frac {\sqrt [4]{-a} \left (\sqrt {b} c-\sqrt {-a} d\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right ) e^{7/2}}{2 b^{3/2} \sqrt {\sqrt {b} c+\sqrt {-a} d} \left (b c^2+a d^2\right )}-\frac {\left (3 b c^2-2 \sqrt {-a} \sqrt {b} d c-2 a d^2\right ) \sqrt {e x} \sqrt {c+d x} e^3}{4 a b d^2 \left (\sqrt {b} c-\sqrt {-a} d\right )^2}-\frac {\left (3 b c^2+2 \sqrt {-a} \sqrt {b} d c-2 a d^2\right ) \sqrt {e x} \sqrt {c+d x} e^3}{4 a b d^2 \left (\sqrt {b} c+\sqrt {-a} d\right )^2}-\frac {\left (\sqrt {b} c+\sqrt {-a} d\right ) \left (3 \sqrt {b} c+4 \sqrt {-a} d\right ) \sqrt {e x} \sqrt {c+d x} e^3}{16 a b d^2 \left (b c^2+a d^2\right )}-\frac {\left (3 b c^2-7 \sqrt {-a} \sqrt {b} d c-4 a d^2\right ) \sqrt {e x} \sqrt {c+d x} e^3}{16 a b d^2 \left (b c^2+a d^2\right )}+\frac {15 c^2 \sqrt {e x} \sqrt {c+d x} e^3}{8 a d^2 \left (b c^2+a d^2\right )}+\frac {\left (\frac {c}{d}+\frac {\sqrt {-a}}{\sqrt {b}}\right ) (e x)^{3/2} \sqrt {c+d x} e^2}{8 a \left (b c^2+a d^2\right )}+\frac {\left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) (e x)^{3/2} \sqrt {c+d x} e^2}{8 a d \left (b c^2+a d^2\right )}-\frac {5 c (e x)^{3/2} \sqrt {c+d x} e^2}{4 a d \left (b c^2+a d^2\right )}+\frac {c \left (2 \sqrt {b} c+\sqrt {-a} d\right ) (e x)^{3/2} e^2}{4 a \sqrt {b} d \left (\sqrt {b} c-\sqrt {-a} d\right )^2 \sqrt {c+d x}}+\frac {c \left (2 \sqrt {b} c-\sqrt {-a} d\right ) (e x)^{3/2} e^2}{4 a \sqrt {b} d \left (\sqrt {b} c+\sqrt {-a} d\right )^2 \sqrt {c+d x}}+\frac {c (e x)^{5/2} e}{a \left (b c^2+a d^2\right ) \sqrt {c+d x}}+\frac {(e x)^{5/2} e}{4 \sqrt {b} \left (\sqrt {-a} \sqrt {b} c-a d\right ) \left (\sqrt {-a}-\sqrt {b} x\right ) \sqrt {c+d x}}-\frac {\sqrt {-a} (e x)^{5/2} e}{4 \sqrt {b} \left (d (-a)^{3/2}+a \sqrt {b} c\right ) \left (\sqrt {b} x+\sqrt {-a}\right ) \sqrt {c+d x}}\)

Input:

Int[(e*x)^(7/2)/((c + d*x)^(3/2)*(a + b*x^2)^2),x]
 

Output:

(c*(2*Sqrt[b]*c - Sqrt[-a]*d)*e^2*(e*x)^(3/2))/(4*a*Sqrt[b]*d*(Sqrt[b]*c + 
 Sqrt[-a]*d)^2*Sqrt[c + d*x]) + (c*(2*Sqrt[b]*c + Sqrt[-a]*d)*e^2*(e*x)^(3 
/2))/(4*a*Sqrt[b]*d*(Sqrt[b]*c - Sqrt[-a]*d)^2*Sqrt[c + d*x]) + (c*e*(e*x) 
^(5/2))/(a*(b*c^2 + a*d^2)*Sqrt[c + d*x]) + (e*(e*x)^(5/2))/(4*Sqrt[b]*(Sq 
rt[-a]*Sqrt[b]*c - a*d)*(Sqrt[-a] - Sqrt[b]*x)*Sqrt[c + d*x]) - (Sqrt[-a]* 
e*(e*x)^(5/2))/(4*Sqrt[b]*(a*Sqrt[b]*c + (-a)^(3/2)*d)*(Sqrt[-a] + Sqrt[b] 
*x)*Sqrt[c + d*x]) - ((3*b*c^2 - 2*Sqrt[-a]*Sqrt[b]*c*d - 2*a*d^2)*e^3*Sqr 
t[e*x]*Sqrt[c + d*x])/(4*a*b*d^2*(Sqrt[b]*c - Sqrt[-a]*d)^2) - ((3*b*c^2 + 
 2*Sqrt[-a]*Sqrt[b]*c*d - 2*a*d^2)*e^3*Sqrt[e*x]*Sqrt[c + d*x])/(4*a*b*d^2 
*(Sqrt[b]*c + Sqrt[-a]*d)^2) + (15*c^2*e^3*Sqrt[e*x]*Sqrt[c + d*x])/(8*a*d 
^2*(b*c^2 + a*d^2)) - ((Sqrt[b]*c + Sqrt[-a]*d)*(3*Sqrt[b]*c + 4*Sqrt[-a]* 
d)*e^3*Sqrt[e*x]*Sqrt[c + d*x])/(16*a*b*d^2*(b*c^2 + a*d^2)) - ((3*b*c^2 - 
 7*Sqrt[-a]*Sqrt[b]*c*d - 4*a*d^2)*e^3*Sqrt[e*x]*Sqrt[c + d*x])/(16*a*b*d^ 
2*(b*c^2 + a*d^2)) + ((Sqrt[-a]/Sqrt[b] + c/d)*e^2*(e*x)^(3/2)*Sqrt[c + d* 
x])/(8*a*(b*c^2 + a*d^2)) - (5*c*e^2*(e*x)^(3/2)*Sqrt[c + d*x])/(4*a*d*(b* 
c^2 + a*d^2)) + ((c - (Sqrt[-a]*d)/Sqrt[b])*e^2*(e*x)^(3/2)*Sqrt[c + d*x]) 
/(8*a*d*(b*c^2 + a*d^2)) - ((-a)^(1/4)*(7*Sqrt[b]*c - 4*Sqrt[-a]*d)*e^(7/2 
)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt 
[c + d*x])])/(4*b^(3/2)*(Sqrt[b]*c - Sqrt[-a]*d)^(5/2)) + ((-a)^(1/4)*(Sqr 
t[b]*c + Sqrt[-a]*d)*e^(7/2)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[...
 

Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(8005\) vs. \(2(416)=832\).

Time = 0.54 (sec) , antiderivative size = 8006, normalized size of antiderivative = 15.70

method	result	size
default	\(\text {Expression too large to display}\)	\(8006\)

Input:

int((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5861 vs. \(2 (416) = 832\).

Time = 1.91 (sec) , antiderivative size = 5861, normalized size of antiderivative = 11.49 \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(7/2)/(d*x+c)**(3/2)/(b*x**2+a)**2,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{2} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate((e*x)^(7/2)/((b*x^2 + a)^2*(d*x + c)^(3/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}}{{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

int((e*x)^(7/2)/((a + b*x^2)^2*(c + d*x)^(3/2)),x)
 

Output:

int((e*x)^(7/2)/((a + b*x^2)^2*(c + d*x)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{7/2}}{(c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, x^{3}}{\sqrt {d x +c}\, a^{2} c +\sqrt {d x +c}\, a^{2} d x +2 \sqrt {d x +c}\, a b c \,x^{2}+2 \sqrt {d x +c}\, a b d \,x^{3}+\sqrt {d x +c}\, b^{2} c \,x^{4}+\sqrt {d x +c}\, b^{2} d \,x^{5}}d x \right ) e^{3} \] Input:

int((e*x)^(7/2)/(d*x+c)^(3/2)/(b*x^2+a)^2,x)
 

Output:

sqrt(e)*int((sqrt(x)*x**3)/(sqrt(c + d*x)*a**2*c + sqrt(c + d*x)*a**2*d*x 
+ 2*sqrt(c + d*x)*a*b*c*x**2 + 2*sqrt(c + d*x)*a*b*d*x**3 + sqrt(c + d*x)* 
b**2*c*x**4 + sqrt(c + d*x)*b**2*d*x**5),x)*e**3