\(\int \frac {(c+d x)^{3/2}}{(e x)^{7/2} (a+b x^2)^2} \, dx\) [911]

Optimal result

Integrand size = 26, antiderivative size = 378 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {9 c \sqrt {c+d x}}{10 a^2 e (e x)^{5/2}}-\frac {13 d \sqrt {c+d x}}{10 a^2 e^2 (e x)^{3/2}}+\frac {\left (45 b c^2-4 a d^2\right ) \sqrt {c+d x}}{10 a^3 c e^3 \sqrt {e x}}+\frac {(c+d x)^{3/2}}{2 a e (e x)^{5/2} \left (a+b x^2\right )}-\frac {3 \sqrt {b} \left (3 b c^2-5 \sqrt {-a} \sqrt {b} c d-2 a d^2\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{13/4} \sqrt {\sqrt {b} c-\sqrt {-a} d} e^{7/2}}+\frac {3 \sqrt {b} \left (3 b c^2+5 \sqrt {-a} \sqrt {b} c d-2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{13/4} \sqrt {\sqrt {b} c+\sqrt {-a} d} e^{7/2}} \] Output:

-9/10*c*(d*x+c)^(1/2)/a^2/e/(e*x)^(5/2)-13/10*d*(d*x+c)^(1/2)/a^2/e^2/(e*x 
)^(3/2)+1/10*(-4*a*d^2+45*b*c^2)*(d*x+c)^(1/2)/a^3/c/e^3/(e*x)^(1/2)+1/2*( 
d*x+c)^(3/2)/a/e/(e*x)^(5/2)/(b*x^2+a)-3/4*b^(1/2)*(3*b*c^2-5*(-a)^(1/2)*b 
^(1/2)*c*d-2*a*d^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)^(13/4)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)/e 
^(7/2)+3/4*b^(1/2)*(3*b*c^2+5*(-a)^(1/2)*b^(1/2)*c*d-2*a*d^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)^(13/4)/(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)/e^(7/2)
 

Mathematica [C] (verified)

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

Time = 1.08 (sec) , antiderivative size = 896, normalized size of antiderivative = 2.37 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )^2} \, dx=\frac {x \left (2 \sqrt {c+d x} \left (45 b^2 c^2 x^4-4 a^2 (c+d x)^2+a b x^2 \left (36 c^2-13 c d x-4 d^2 x^2\right )\right )+5 c \sqrt {d} x^{5/2} \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 {b^2 c^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-66 a b c^2 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+64 a^2 d^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-2 b^2 c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-24 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}+b^2 c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2-2 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 ]+20 c \sqrt {d} x^{5/2} \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 )-4 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 ]\right )}{20 a^3 c (e x)^{7/2} \left (a+b x^2\right )} \] Input:

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

Output:

(x*(2*Sqrt[c + d*x]*(45*b^2*c^2*x^4 - 4*a^2*(c + d*x)^2 + a*b*x^2*(36*c^2 
- 13*c*d*x - 4*d^2*x^2)) + 5*c*Sqrt[d]*x^(5/2)*(a + b*x^2)*RootSum[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 & , (b^2* 
c^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 66*a*b*c^2*d^2 
*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 64*a^2*d^4*Log[c 
+ 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 2*b^2*c^3*Log[c + 2*d*x 
- 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - 24*a*b*c*d^2*Log[c + 2*d*x - 
2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + b^2*c^2*Log[c + 2*d*x - 2*Sqrt[ 
d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2 - 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) & ] + 20*c*Sqrt[d]*x^(5/2)*(a + b*x^2)*RootSum[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*L 
og[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] - 4*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*Log[c + 2*d*x - 2*S 
qrt[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]*Sqr 
t[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) & ]))/(20*a^3*c*(e*x)^(7/2)*(a + b*x^2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1236\) vs. \(2(378)=756\).

Time = 5.74 (sec) , antiderivative size = 1236, normalized size of antiderivative = 3.27, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 \sqrt {c+d x} d^2}{5 a^2 c e^3 \sqrt {e x}}-\frac {2 \sqrt {c+d x} d}{5 a^2 e^2 (e x)^{3/2}}-\frac {\sqrt {b} \left (b c^2-2 \sqrt {-a} \sqrt {b} d c-a d^2\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{2 (-a)^{13/4} \sqrt {\sqrt {b} c-\sqrt {-a} d} e^{7/2}}-\frac {\sqrt {b} \left (7 b^{3/2} c^3-18 \sqrt {-a} b d c^2-15 a \sqrt {b} d^2 c+4 \sqrt {-a} a d^3\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{13/4} \left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2} e^{7/2}}+\frac {\sqrt {b} \left (b c^2+2 \sqrt {-a} \sqrt {b} d c-a d^2\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 )}{2 (-a)^{13/4} \sqrt {\sqrt {b} c+\sqrt {-a} d} e^{7/2}}+\frac {\sqrt {b} \left (7 b^{3/2} c^3+18 \sqrt {-a} b d c^2-15 a \sqrt {b} d^2 c+4 (-a)^{3/2} d^3\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 )}{4 (-a)^{13/4} \left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2} e^{7/2}}+\frac {\left (b c^2-a d^2\right ) \sqrt {c+d x}}{a^3 c e^3 \sqrt {e x}}-\frac {\left (105 b^{3/2} c^3+200 \sqrt {-a} b d c^2-101 a \sqrt {b} d^2 c-6 \sqrt {-a} a d^3\right ) \sqrt {c+d x}}{60 (-a)^{5/2} c \left (\sqrt {-a} \sqrt {b} c-a d\right ) e^3 \sqrt {e x}}-\frac {\left (105 b^{3/2} c^3-200 \sqrt {-a} b d c^2-101 a \sqrt {b} d^2 c+6 \sqrt {-a} a d^3\right ) \sqrt {c+d x}}{60 (-a)^{5/2} c \left (\sqrt {-a} \sqrt {b} c+a d\right ) e^3 \sqrt {e x}}+\frac {\left (7 \sqrt {b} c+5 \sqrt {-a} d\right ) \sqrt {c+d x}}{20 a^2 e^2 (e x)^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {c \sqrt {c+d x}}{10 (-a)^{3/2} e (e x)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\left (7 \sqrt {b} c-5 \sqrt {-a} d\right ) \sqrt {c+d x}}{20 a^2 e^2 (e x)^{3/2} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {c \sqrt {c+d x}}{10 (-a)^{3/2} e (e x)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {\left (35 b c^2-62 \sqrt {-a} \sqrt {b} d c-27 a d^2\right ) \sqrt {c+d x}}{60 a^2 \left (\sqrt {-a} \sqrt {b} c+a d\right ) e^2 (e x)^{3/2}}-\frac {\left (35 b c^2+62 \sqrt {-a} \sqrt {b} d c-27 a d^2\right ) \sqrt {c+d x}}{60 a^2 \left (\sqrt {-a} \sqrt {b} c-a d\right ) e^2 (e x)^{3/2}}-\frac {c \sqrt {c+d x}}{5 a^2 e (e x)^{5/2}}\)

Input:

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

Output:

-1/5*(c*Sqrt[c + d*x])/(a^2*e*(e*x)^(5/2)) - (2*d*Sqrt[c + d*x])/(5*a^2*e^ 
2*(e*x)^(3/2)) + ((35*b*c^2 - 62*Sqrt[-a]*Sqrt[b]*c*d - 27*a*d^2)*Sqrt[c + 
 d*x])/(60*a^2*(Sqrt[-a]*Sqrt[b]*c + a*d)*e^2*(e*x)^(3/2)) - ((35*b*c^2 + 
62*Sqrt[-a]*Sqrt[b]*c*d - 27*a*d^2)*Sqrt[c + d*x])/(60*a^2*(Sqrt[-a]*Sqrt[ 
b]*c - a*d)*e^2*(e*x)^(3/2)) + (4*d^2*Sqrt[c + d*x])/(5*a^2*c*e^3*Sqrt[e*x 
]) + ((b*c^2 - a*d^2)*Sqrt[c + d*x])/(a^3*c*e^3*Sqrt[e*x]) - ((105*b^(3/2) 
*c^3 + 200*Sqrt[-a]*b*c^2*d - 101*a*Sqrt[b]*c*d^2 - 6*Sqrt[-a]*a*d^3)*Sqrt 
[c + d*x])/(60*(-a)^(5/2)*c*(Sqrt[-a]*Sqrt[b]*c - a*d)*e^3*Sqrt[e*x]) - (( 
105*b^(3/2)*c^3 - 200*Sqrt[-a]*b*c^2*d - 101*a*Sqrt[b]*c*d^2 + 6*Sqrt[-a]* 
a*d^3)*Sqrt[c + d*x])/(60*(-a)^(5/2)*c*(Sqrt[-a]*Sqrt[b]*c + a*d)*e^3*Sqrt 
[e*x]) - (c*Sqrt[c + d*x])/(10*(-a)^(3/2)*e*(e*x)^(5/2)*(Sqrt[-a] - Sqrt[b 
]*x)) + ((7*Sqrt[b]*c + 5*Sqrt[-a]*d)*Sqrt[c + d*x])/(20*a^2*e^2*(e*x)^(3/ 
2)*(Sqrt[-a] - Sqrt[b]*x)) - (c*Sqrt[c + d*x])/(10*(-a)^(3/2)*e*(e*x)^(5/2 
)*(Sqrt[-a] + Sqrt[b]*x)) - ((7*Sqrt[b]*c - 5*Sqrt[-a]*d)*Sqrt[c + d*x])/( 
20*a^2*e^2*(e*x)^(3/2)*(Sqrt[-a] + Sqrt[b]*x)) - (Sqrt[b]*(b*c^2 - 2*Sqrt[ 
-a]*Sqrt[b]*c*d - a*d^2)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/( 
(-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(2*(-a)^(13/4)*Sqrt[Sqrt[b]*c - Sqrt[- 
a]*d]*e^(7/2)) - (Sqrt[b]*(7*b^(3/2)*c^3 - 18*Sqrt[-a]*b*c^2*d - 15*a*Sqrt 
[b]*c*d^2 + 4*Sqrt[-a]*a*d^3)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e* 
x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/(4*(-a)^(13/4)*(Sqrt[b]*c - Sq...
 

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. \(3212\) vs. \(2(290)=580\).

Time = 0.51 (sec) , antiderivative size = 3213, normalized size of antiderivative = 8.50

method	result	size
default	\(\text {Expression too large to display}\)	\(3213\)
risch	\(\text {Expression too large to display}\)	\(3547\)

Input:

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

Output:

-1/40*(d*x+c)^(1/2)*b*(-30*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*ln((2*(-a*b)^ 
(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2 
)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b)^(1/2)))*a^3*(-a*b)^(1/2)*c*d^4*e*x^3-30* 
ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(-e*(a*d+c*(-a*b)^ 
(1/2))/b)^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)))*(e*(-a*d+c*(-a*b)^ 
(1/2))/b)^(1/2)*a^3*(-a*b)^(1/2)*c*d^4*e*x^3+16*a^3*d^4*x^2*((d*x+c)*e*x)^ 
(1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)* 
(-a*b)^(1/2)+16*a^3*c^2*d^2*((d*x+c)*e*x)^(1/2)*(-e*(a*d+c*(-a*b)^(1/2))/b 
)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*(-a*b)^(1/2)+45*(e*(-a*d+c*(-a*b 
)^(1/2))/b)^(1/2)*ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)* 
(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)))* 
a*b^2*(-a*b)^(1/2)*c^5*e*x^3-75*ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c 
)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(- 
a*b)^(1/2)))*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*a^3*b*c^2*d^3*e*x^3-75*ln(( 
2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2) 
)/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b)^(1/2)))*(-e*(a*d+c*(-a*b)^(1/2) 
)/b)^(1/2)*a^2*b^2*c^4*d*e*x^3+45*ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x 
+c)*e*x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x- 
(-a*b)^(1/2)))*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*a*b^2*(-a*b)^(1/2)*c^5*e* 
x^3-75*ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(e*(-a*d+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1413 vs. \(2 (290) = 580\).

Time = 0.12 (sec) , antiderivative size = 1413, normalized size of antiderivative = 3.74 \[ \int \frac {(c+d x)^{3/2}}{(e x)^{7/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/40*(15*(a^3*b*c*e^4*x^5 + a^4*c*e^4*x^3)*sqrt((a^6*e^7*sqrt(-(81*b^5*c^ 
6 - 288*a*b^4*c^4*d^2 + 256*a^2*b^3*c^2*d^4)/(a^13*e^14)) + 21*b^2*c^2*d - 
 4*a*b*d^3)/(a^6*e^7))*log(-27*((81*b^4*c^5 - 108*a*b^3*c^3*d^2 - 64*a^2*b 
^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) + (2*a^10*d*e^11*x*sqrt(-(81*b^5*c^6 - 2 
88*a*b^4*c^4*d^2 + 256*a^2*b^3*c^2*d^4)/(a^13*e^14)) + 3*(9*a^3*b^3*c^4 - 
16*a^4*b^2*c^2*d^2)*e^4*x)*sqrt((a^6*e^7*sqrt(-(81*b^5*c^6 - 288*a*b^4*c^4 
*d^2 + 256*a^2*b^3*c^2*d^4)/(a^13*e^14)) + 21*b^2*c^2*d - 4*a*b*d^3)/(a^6* 
e^7)))/x) - 15*(a^3*b*c*e^4*x^5 + a^4*c*e^4*x^3)*sqrt((a^6*e^7*sqrt(-(81*b 
^5*c^6 - 288*a*b^4*c^4*d^2 + 256*a^2*b^3*c^2*d^4)/(a^13*e^14)) + 21*b^2*c^ 
2*d - 4*a*b*d^3)/(a^6*e^7))*log(-27*((81*b^4*c^5 - 108*a*b^3*c^3*d^2 - 64* 
a^2*b^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) - (2*a^10*d*e^11*x*sqrt(-(81*b^5*c^ 
6 - 288*a*b^4*c^4*d^2 + 256*a^2*b^3*c^2*d^4)/(a^13*e^14)) + 3*(9*a^3*b^3*c 
^4 - 16*a^4*b^2*c^2*d^2)*e^4*x)*sqrt((a^6*e^7*sqrt(-(81*b^5*c^6 - 288*a*b^ 
4*c^4*d^2 + 256*a^2*b^3*c^2*d^4)/(a^13*e^14)) + 21*b^2*c^2*d - 4*a*b*d^3)/ 
(a^6*e^7)))/x) - 15*(a^3*b*c*e^4*x^5 + a^4*c*e^4*x^3)*sqrt(-(a^6*e^7*sqrt( 
-(81*b^5*c^6 - 288*a*b^4*c^4*d^2 + 256*a^2*b^3*c^2*d^4)/(a^13*e^14)) - 21* 
b^2*c^2*d + 4*a*b*d^3)/(a^6*e^7))*log(-27*((81*b^4*c^5 - 108*a*b^3*c^3*d^2 
 - 64*a^2*b^2*c*d^4)*sqrt(d*x + c)*sqrt(e*x) + (2*a^10*d*e^11*x*sqrt(-(81* 
b^5*c^6 - 288*a*b^4*c^4*d^2 + 256*a^2*b^3*c^2*d^4)/(a^13*e^14)) - 3*(9*a^3 
*b^3*c^4 - 16*a^4*b^2*c^2*d^2)*e^4*x)*sqrt(-(a^6*e^7*sqrt(-(81*b^5*c^6 ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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