3.780 \(\int \frac{1}{x^3 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)
Optimal. Leaf size=361 \[ \frac{7 a d+5 b c}{4 a^2 c^2 x \sqrt{a+b x} (c+d x)^{3/2}}-\frac{5 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} c^{9/2}}+\frac{b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}+\frac{d \sqrt{a+b x} \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )}{12 a^3 c^3 (c+d x)^{3/2} (b c-a d)^2}+\frac{d \sqrt{a+b x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{12 a^3 c^4 \sqrt{c+d x} (b c-a d)^3}-\frac{1}{2 a c x^2 \sqrt{a+b x} (c+d x)^{3/2}} \]
[Out]
(b*(15*b^2*c^2 - 7*a^2*d^2))/(4*a^3*c^2*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2
)) - 1/(2*a*c*x^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (5*b*c + 7*a*d)/(4*a^2*c^2*x*
Sqrt[a + b*x]*(c + d*x)^(3/2)) + (d*(45*b^3*c^3 - 15*a*b^2*c^2*d - 33*a^2*b*c*d^
2 + 35*a^3*d^3)*Sqrt[a + b*x])/(12*a^3*c^3*(b*c - a*d)^2*(c + d*x)^(3/2)) + (d*(
45*b^4*c^4 - 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 105*a^4*d^4
)*Sqrt[a + b*x])/(12*a^3*c^4*(b*c - a*d)^3*Sqrt[c + d*x]) - (5*(3*b^2*c^2 + 6*a*
b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*
a^(7/2)*c^(9/2))
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Rubi [A] time = 1.24787, antiderivative size = 361, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273
\[ \frac{7 a d+5 b c}{4 a^2 c^2 x \sqrt{a+b x} (c+d x)^{3/2}}-\frac{5 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} c^{9/2}}+\frac{b \left (15 b^2 c^2-7 a^2 d^2\right )}{4 a^3 c^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}+\frac{d \sqrt{a+b x} \left (35 a^3 d^3-33 a^2 b c d^2-15 a b^2 c^2 d+45 b^3 c^3\right )}{12 a^3 c^3 (c+d x)^{3/2} (b c-a d)^2}+\frac{d \sqrt{a+b x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{12 a^3 c^4 \sqrt{c+d x} (b c-a d)^3}-\frac{1}{2 a c x^2 \sqrt{a+b x} (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
(b*(15*b^2*c^2 - 7*a^2*d^2))/(4*a^3*c^2*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2
)) - 1/(2*a*c*x^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (5*b*c + 7*a*d)/(4*a^2*c^2*x*
Sqrt[a + b*x]*(c + d*x)^(3/2)) + (d*(45*b^3*c^3 - 15*a*b^2*c^2*d - 33*a^2*b*c*d^
2 + 35*a^3*d^3)*Sqrt[a + b*x])/(12*a^3*c^3*(b*c - a*d)^2*(c + d*x)^(3/2)) + (d*(
45*b^4*c^4 - 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 105*a^4*d^4
)*Sqrt[a + b*x])/(12*a^3*c^4*(b*c - a*d)^3*Sqrt[c + d*x]) - (5*(3*b^2*c^2 + 6*a*
b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*
a^(7/2)*c^(9/2))
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
Timed out
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Mathematica [A] time = 1.6709, size = 258, normalized size = 0.71 \[ \frac{5 \log (x) \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right )}{8 a^{7/2} c^{9/2}}-\frac{5 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{7/2} c^{9/2}}+\frac{1}{12} \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{24 b^5}{a^3 (a+b x) (a d-b c)^3}+\frac{33 a d+21 b c}{a^3 c^4 x}-\frac{6}{a^2 c^3 x^2}+\frac{8 d^4 (14 b c-9 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{8 d^4}{c^3 (c+d x)^2 (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(-6/(a^2*c^3*x^2) + (21*b*c + 33*a*d)/(a^3*c^4*x) -
(24*b^5)/(a^3*(-(b*c) + a*d)^3*(a + b*x)) + (8*d^4)/(c^3*(b*c - a*d)^2*(c + d*x
)^2) + (8*d^4*(14*b*c - 9*a*d))/(c^4*(b*c - a*d)^3*(c + d*x))))/12 + (5*(3*b^2*c
^2 + 6*a*b*c*d + 7*a^2*d^2)*Log[x])/(8*a^(7/2)*c^(9/2)) - (5*(3*b^2*c^2 + 6*a*b*
c*d + 7*a^2*d^2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqr
t[c + d*x]])/(8*a^(7/2)*c^(9/2))
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Maple [B] time = 0.08, size = 2216, normalized size = 6.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x)
[Out]
-1/24*(30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^
2*b^4*c^3*d^4+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x
)*x^4*a^6*d^7-45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)
*x^3*b^6*c^7-210*x^3*a^5*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-225*ln((a*d*x+b
*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^4*b^2*c*d^6+90*ln((a*
d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^3*b^3*c^2*d^5+45
*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a*b^5*c^4*d
^3-15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^5*b*
c*d^6-360*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^
4*b^2*c^2*d^5+210*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x
)*x^4*a^3*b^3*c^3*d^4+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+
2*a*c)/x)*x^4*a^2*b^4*c^4*d^3+45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)+2*a*c)/x)*x^4*a*b^5*c^5*d^2-345*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)+2*a*c)/x)*x^3*a^5*b*c^2*d^5-45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+
a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^4*b^2*c^3*d^4+150*ln((a*d*x+b*c*x+2*(a*c)^(1/2
)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^3*b^3*c^4*d^3+120*ln((a*d*x+b*c*x+2*(a
*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^2*b^4*c^5*d^2-45*ln((a*d*x+b*c
*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a*b^5*c^6*d-225*ln((a*d*x
+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a^5*b*c^3*d^4+90*ln((
a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a^4*b^2*c^4*d^3+
30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a^3*b^3*c
^5*d^2+45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a^
2*b^4*c^6*d-210*x^4*a^4*b*d^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+90*x^4*b^5*c^4
*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+180*x^3*b^5*c^5*d*(a*c)^(1/2)*((b*x+a)*
(d*x+c))^(1/2)-280*x^2*a^5*c*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-42*x*a^5*c^
2*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+30*x*a*b^4*c^6*(a*c)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)-36*a^4*b*c^4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+36*a^3*b^2*c^5
*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+380*x^4*a^3*b^2*c*d^5*(a*c)^(1/2)*((b*x+a
)*(d*x+c))^(1/2)-72*x^4*a^2*b^3*c^2*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-60*x
^4*a*b^4*c^3*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+100*x^3*a^4*b*c*d^5*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)+444*x^3*a^3*b^2*c^2*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)-168*x^3*a^2*b^3*c^3*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-90*x^3*a*b^
4*c^4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+474*x^2*a^4*b*c^2*d^4*(a*c)^(1/2)*
((b*x+a)*(d*x+c))^(1/2)-24*x^2*a^3*b^2*c^3*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)-132*x^2*a^2*b^3*c^4*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+96*x*a^4*b*c^3*d^
3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-36*x*a^3*b^2*c^4*d^2*(a*c)^(1/2)*((b*x+a)*
(d*x+c))^(1/2)-48*x*a^2*b^3*c^5*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+90*x^2*b^5
*c^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*a^5*c^3*d^3*(a*c)^(1/2)*((b*x+a)*(d*
x+c))^(1/2)-12*a^2*b^3*c^6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+105*ln((a*d*x+b*c
*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^5*b*d^7-45*ln((a*d*x+b*
c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*b^6*c^5*d^2-90*ln((a*d*x
+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*b^6*c^6*d+210*ln((a*d
*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^6*c*d^6+105*ln((a
*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a^6*c^2*d^5-45*ln
((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^2*a*b^5*c^7)/c^4
/a^3/x^2/(a*c)^(1/2)/(a*d-b*c)^3/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)/(b*x+a)^(
1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{5}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x^3),x, algorithm="maxima")
[Out]
integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x^3), x)
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Fricas [A] time = 1.66737, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x^3),x, algorithm="fricas")
[Out]
[-1/48*(4*(6*a^2*b^3*c^6 - 18*a^3*b^2*c^5*d + 18*a^4*b*c^4*d^2 - 6*a^5*c^3*d^3 -
(45*b^5*c^4*d^2 - 30*a*b^4*c^3*d^3 - 36*a^2*b^3*c^2*d^4 + 190*a^3*b^2*c*d^5 - 1
05*a^4*b*d^6)*x^4 - (90*b^5*c^5*d - 45*a*b^4*c^4*d^2 - 84*a^2*b^3*c^3*d^3 + 222*
a^3*b^2*c^2*d^4 + 50*a^4*b*c*d^5 - 105*a^5*d^6)*x^3 - (45*b^5*c^6 - 66*a^2*b^3*c
^4*d^2 - 12*a^3*b^2*c^3*d^3 + 237*a^4*b*c^2*d^4 - 140*a^5*c*d^5)*x^2 - 3*(5*a*b^
4*c^6 - 8*a^2*b^3*c^5*d - 6*a^3*b^2*c^4*d^2 + 16*a^4*b*c^3*d^3 - 7*a^5*c^2*d^4)*
x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 15*((3*b^6*c^5*d^2 - 3*a*b^5*c^4*d^3
- 2*a^2*b^4*c^3*d^4 - 6*a^3*b^3*c^2*d^5 + 15*a^4*b^2*c*d^6 - 7*a^5*b*d^7)*x^5 +
(6*b^6*c^6*d - 3*a*b^5*c^5*d^2 - 7*a^2*b^4*c^4*d^3 - 14*a^3*b^3*c^3*d^4 + 24*a^4
*b^2*c^2*d^5 + a^5*b*c*d^6 - 7*a^6*d^7)*x^4 + (3*b^6*c^7 + 3*a*b^5*c^6*d - 8*a^2
*b^4*c^5*d^2 - 10*a^3*b^3*c^4*d^3 + 3*a^4*b^2*c^3*d^4 + 23*a^5*b*c^2*d^5 - 14*a^
6*c*d^6)*x^3 + (3*a*b^5*c^7 - 3*a^2*b^4*c^6*d - 2*a^3*b^3*c^5*d^2 - 6*a^4*b^2*c^
4*d^3 + 15*a^5*b*c^3*d^4 - 7*a^6*c^2*d^5)*x^2)*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a
^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) - (8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2
*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2))/(((a^3*b^4*c^7*d^2 - 3*a^4
*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^5 + (2*a^3*b^4*c^8*d - 5*a^4
*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^4 + (a^3*b^4*c
^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^3 +
(a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2)*sqrt(a*c)),
-1/24*(2*(6*a^2*b^3*c^6 - 18*a^3*b^2*c^5*d + 18*a^4*b*c^4*d^2 - 6*a^5*c^3*d^3 -
(45*b^5*c^4*d^2 - 30*a*b^4*c^3*d^3 - 36*a^2*b^3*c^2*d^4 + 190*a^3*b^2*c*d^5 - 1
05*a^4*b*d^6)*x^4 - (90*b^5*c^5*d - 45*a*b^4*c^4*d^2 - 84*a^2*b^3*c^3*d^3 + 222*
a^3*b^2*c^2*d^4 + 50*a^4*b*c*d^5 - 105*a^5*d^6)*x^3 - (45*b^5*c^6 - 66*a^2*b^3*c
^4*d^2 - 12*a^3*b^2*c^3*d^3 + 237*a^4*b*c^2*d^4 - 140*a^5*c*d^5)*x^2 - 3*(5*a*b^
4*c^6 - 8*a^2*b^3*c^5*d - 6*a^3*b^2*c^4*d^2 + 16*a^4*b*c^3*d^3 - 7*a^5*c^2*d^4)*
x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((3*b^6*c^5*d^2 - 3*a*b^5*c^4*d^3
- 2*a^2*b^4*c^3*d^4 - 6*a^3*b^3*c^2*d^5 + 15*a^4*b^2*c*d^6 - 7*a^5*b*d^7)*x^5 +
(6*b^6*c^6*d - 3*a*b^5*c^5*d^2 - 7*a^2*b^4*c^4*d^3 - 14*a^3*b^3*c^3*d^4 + 24*a^
4*b^2*c^2*d^5 + a^5*b*c*d^6 - 7*a^6*d^7)*x^4 + (3*b^6*c^7 + 3*a*b^5*c^6*d - 8*a^
2*b^4*c^5*d^2 - 10*a^3*b^3*c^4*d^3 + 3*a^4*b^2*c^3*d^4 + 23*a^5*b*c^2*d^5 - 14*a
^6*c*d^6)*x^3 + (3*a*b^5*c^7 - 3*a^2*b^4*c^6*d - 2*a^3*b^3*c^5*d^2 - 6*a^4*b^2*c
^4*d^3 + 15*a^5*b*c^3*d^4 - 7*a^6*c^2*d^5)*x^2)*arctan(1/2*(2*a*c + (b*c + a*d)*
x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/(((a^3*b^4*c^7*d^2 - 3*a^4*b^3
*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^5 + (2*a^3*b^4*c^8*d - 5*a^4*b^3
*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^4 + (a^3*b^4*c^9 -
a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^3 + (a^4
*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2)*sqrt(-a*c))]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 2.41185, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x^3),x, algorithm="giac")
[Out]
sage0*x