3.772 \(\int \frac{x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=351 \[ -\frac{2 c x^2 \sqrt{a+b x} \left (-3 a^2 d^2-12 a b c d+7 b^2 c^2\right )}{3 b d^2 \sqrt{c+d x} (b c-a d)^3}+\frac{5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{7/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-2 b d x \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )-190 a b^3 c^3 d+105 b^4 c^4\right )}{12 b^3 d^4 (b c-a d)^3}+\frac{2 a x^4}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}-\frac{2 c x^3 \sqrt{a+b x} (3 a d+b c)}{3 b d (c+d x)^{3/2} (b c-a d)^2} \]

[Out]

(2*a*x^4)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b*c + 3*a*d)*x^3
*Sqrt[a + b*x])/(3*b*d*(b*c - a*d)^2*(c + d*x)^(3/2)) - (2*c*(7*b^2*c^2 - 12*a*b
*c*d - 3*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*b*d^2*(b*c - a*d)^3*Sqrt[c + d*x]) - (Sq
rt[a + b*x]*Sqrt[c + d*x]*(105*b^4*c^4 - 190*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 +
30*a^3*b*c*d^3 - 45*a^4*d^4 - 2*b*d*(35*b^3*c^3 - 61*a*b^2*c^2*d + 9*a^2*b*c*d^2
- 15*a^3*d^3)*x))/(12*b^3*d^4*(b*c - a*d)^3) + (5*(7*b^2*c^2 + 6*a*b*c*d + 3*a^
2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(7/2)*d^(9
/2))

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Rubi [A]  time = 0.924338, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 c x^2 \sqrt{a+b x} \left (-3 a^2 d^2-12 a b c d+7 b^2 c^2\right )}{3 b d^2 \sqrt{c+d x} (b c-a d)^3}+\frac{5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{7/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-2 b d x \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )-190 a b^3 c^3 d+105 b^4 c^4\right )}{12 b^3 d^4 (b c-a d)^3}+\frac{2 a x^4}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}-\frac{2 c x^3 \sqrt{a+b x} (3 a d+b c)}{3 b d (c+d x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

(2*a*x^4)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b*c + 3*a*d)*x^3
*Sqrt[a + b*x])/(3*b*d*(b*c - a*d)^2*(c + d*x)^(3/2)) - (2*c*(7*b^2*c^2 - 12*a*b
*c*d - 3*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*b*d^2*(b*c - a*d)^3*Sqrt[c + d*x]) - (Sq
rt[a + b*x]*Sqrt[c + d*x]*(105*b^4*c^4 - 190*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 +
30*a^3*b*c*d^3 - 45*a^4*d^4 - 2*b*d*(35*b^3*c^3 - 61*a*b^2*c^2*d + 9*a^2*b*c*d^2
- 15*a^3*d^3)*x))/(12*b^3*d^4*(b*c - a*d)^3) + (5*(7*b^2*c^2 + 6*a*b*c*d + 3*a^
2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(7/2)*d^(9
/2))

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Rubi in Sympy [A]  time = 91.8243, size = 357, normalized size = 1.02 \[ - \frac{2 a x^{4}}{b \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 c x^{3} \sqrt{a + b x} \left (3 a d + b c\right )}{3 b d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{2 c x^{2} \sqrt{a + b x} \left (3 a^{2} d^{2} + 12 a b c d - 7 b^{2} c^{2}\right )}{3 b d^{2} \sqrt{c + d x} \left (a d - b c\right )^{3}} - \frac{4 \sqrt{a + b x} \sqrt{c + d x} \left (\frac{45 a^{4} d^{4}}{16} - \frac{15 a^{3} b c d^{3}}{8} - \frac{9 a^{2} b^{2} c^{2} d^{2}}{4} + \frac{95 a b^{3} c^{3} d}{8} - \frac{105 b^{4} c^{4}}{16} - \frac{b d x \left (15 a^{3} d^{3} - 9 a^{2} b c d^{2} + 61 a b^{2} c^{2} d - 35 b^{3} c^{3}\right )}{8}\right )}{3 b^{3} d^{4} \left (a d - b c\right )^{3}} + \frac{5 \left (3 a^{2} d^{2} + 6 a b c d + 7 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{7}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

-2*a*x**4/(b*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)) - 2*c*x**3*sqrt(a + b*x
)*(3*a*d + b*c)/(3*b*d*(c + d*x)**(3/2)*(a*d - b*c)**2) - 2*c*x**2*sqrt(a + b*x)
*(3*a**2*d**2 + 12*a*b*c*d - 7*b**2*c**2)/(3*b*d**2*sqrt(c + d*x)*(a*d - b*c)**3
) - 4*sqrt(a + b*x)*sqrt(c + d*x)*(45*a**4*d**4/16 - 15*a**3*b*c*d**3/8 - 9*a**2
*b**2*c**2*d**2/4 + 95*a*b**3*c**3*d/8 - 105*b**4*c**4/16 - b*d*x*(15*a**3*d**3
- 9*a**2*b*c*d**2 + 61*a*b**2*c**2*d - 35*b**3*c**3)/8)/(3*b**3*d**4*(a*d - b*c)
**3) + 5*(3*a**2*d**2 + 6*a*b*c*d + 7*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sq
rt(b)*sqrt(c + d*x)))/(4*b**(7/2)*d**(9/2))

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Mathematica [A]  time = 0.919369, size = 214, normalized size = 0.61 \[ \frac{1}{12} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{24 a^5}{b^3 (a+b x) (b c-a d)^3}-\frac{3 (7 a d+11 b c)}{b^3 d^4}+\frac{8 c^5}{d^4 (c+d x)^2 (b c-a d)^2}+\frac{40 c^4 (2 b c-3 a d)}{d^4 (c+d x) (a d-b c)^3}+\frac{6 x}{b^2 d^3}\right )+\frac{5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{7/2} d^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*((-3*(11*b*c + 7*a*d))/(b^3*d^4) + (6*x)/(b^2*d^3)
+ (24*a^5)/(b^3*(b*c - a*d)^3*(a + b*x)) + (8*c^5)/(d^4*(b*c - a*d)^2*(c + d*x)^
2) + (40*c^4*(2*b*c - 3*a*d))/(d^4*(-(b*c) + a*d)^3*(c + d*x))))/12 + (5*(7*b^2*
c^2 + 6*a*b*c*d + 3*a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a
+ b*x]*Sqrt[c + d*x]])/(8*b^(7/2)*d^(9/2))

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Maple [B]  time = 0.062, size = 2228, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x+a)^(3/2)/(d*x+c)^(5/2),x)

[Out]

1/24*(-150*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^
(1/2))*x^2*a^3*b^3*c^3*d^4+45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^6*d^7-105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*b^6*c^7+45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d
*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^6*c^2*d^5-105*ln(1/2*(2*b*d*x+2
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^5*c^7-90*x^2*a^5*
d^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-474*x^2*a*b^4*c^4*d^2*((b*x+a)*(d*x+c))^
(1/2)*(b*d)^(1/2)+90*x*a^4*b*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+168*x*a
^3*b^2*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-444*x*a^2*b^3*c^4*d^2*((b*x+a
)*(d*x+c))^(1/2)*(b*d)^(1/2)-100*x*a*b^4*c^5*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/
2)-36*x^4*a^2*b^3*c*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+36*x^4*a*b^4*c^2*d^4
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+48*x^3*a^3*b^2*c*d^5*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)+36*x^3*a^2*b^3*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-96*x^3*
a*b^4*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+132*x^2*a^3*b^2*c^2*d^4*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2)+24*x^2*a^2*b^3*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)+225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d
)^(1/2))*x^3*a*b^5*c^4*d^3+45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^5*b*c*d^6-120*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^4*b^2*c^2*d^5-90*ln(1/2*(2*b*d*x
+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*a^2*b^4*c^3*d^4
-30*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*
x^3*a^3*b^3*c^2*d^5-45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d
+b*c)/(b*d)^(1/2))*x^3*a^4*b^2*c*d^6-30*x^3*a^4*b*d^6*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)+42*x^3*b^5*c^4*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+280*x^2*b^5*c^5
*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-180*x*a^5*c*d^5*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)+60*a^4*b*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+72*a^3*b^2*c^4*d
^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-380*a^2*b^3*c^5*d*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)+45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(
b*d)^(1/2))*x^2*a^2*b^4*c^4*d^3+345*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a*b^5*c^5*d^2-45*ln(1/2*(2*b*d*x+2*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^5*b*c^2*d^5-105*ln(1/2*(2*b
*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^4*b^2*c^3*d
^4-210*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2
))*x*a^3*b^3*c^4*d^3+360*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a
*d+b*c)/(b*d)^(1/2))*x*a^2*b^4*c^5*d^2+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a*b^5*c^6*d+12*x^4*a^3*b^2*d^6*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)-12*x^4*b^5*c^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)+210*x*b^5*c^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-90*a^5*c^2*d^4*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)+210*a*b^4*c^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+45*ln(1
/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*a^5*
b*d^7-105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(
1/2))*x^3*b^6*c^5*d^2-210*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+
a*d+b*c)/(b*d)^(1/2))*x^2*b^6*c^6*d+90*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^6*c*d^6-45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b*c^3*d^4-30*ln(1/2*(2*b*d*x+2
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2*c^4*d^3-90*ln
(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^
3*c^5*d^2+225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*
d)^(1/2))*a^2*b^4*c^6*d)/(b*d)^(1/2)/(a*d-b*c)^3/((b*x+a)*(d*x+c))^(1/2)/d^4/b^3
/(d*x+c)^(3/2)/(b*x+a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.4386, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[-1/48*(4*(105*a*b^4*c^6 - 190*a^2*b^3*c^5*d + 36*a^3*b^2*c^4*d^2 + 30*a^4*b*c^3
*d^3 - 45*a^5*c^2*d^4 - 6*(b^5*c^3*d^3 - 3*a*b^4*c^2*d^4 + 3*a^2*b^3*c*d^5 - a^3
*b^2*d^6)*x^4 + 3*(7*b^5*c^4*d^2 - 16*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 + 8*a^3*
b^2*c*d^5 - 5*a^4*b*d^6)*x^3 + (140*b^5*c^5*d - 237*a*b^4*c^4*d^2 + 12*a^2*b^3*c
^3*d^3 + 66*a^3*b^2*c^2*d^4 - 45*a^5*d^6)*x^2 + (105*b^5*c^6 - 50*a*b^4*c^5*d -
222*a^2*b^3*c^4*d^2 + 84*a^3*b^2*c^3*d^3 + 45*a^4*b*c^2*d^4 - 90*a^5*c*d^5)*x)*s
qrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(7*a*b^5*c^7 - 15*a^2*b^4*c^6*d + 6*a^
3*b^3*c^5*d^2 + 2*a^4*b^2*c^4*d^3 + 3*a^5*b*c^3*d^4 - 3*a^6*c^2*d^5 + (7*b^6*c^5
*d^2 - 15*a*b^5*c^4*d^3 + 6*a^2*b^4*c^3*d^4 + 2*a^3*b^3*c^2*d^5 + 3*a^4*b^2*c*d^
6 - 3*a^5*b*d^7)*x^3 + (14*b^6*c^6*d - 23*a*b^5*c^5*d^2 - 3*a^2*b^4*c^4*d^3 + 10
*a^3*b^3*c^3*d^4 + 8*a^4*b^2*c^2*d^5 - 3*a^5*b*c*d^6 - 3*a^6*d^7)*x^2 + (7*b^6*c
^7 - a*b^5*c^6*d - 24*a^2*b^4*c^5*d^2 + 14*a^3*b^3*c^4*d^3 + 7*a^4*b^2*c^3*d^4 +
3*a^5*b*c^2*d^5 - 6*a^6*c*d^6)*x)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(
b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2
*c*d + a*b*d^2)*x)*sqrt(b*d)))/((a*b^6*c^5*d^4 - 3*a^2*b^5*c^4*d^5 + 3*a^3*b^4*c
^3*d^6 - a^4*b^3*c^2*d^7 + (b^7*c^3*d^6 - 3*a*b^6*c^2*d^7 + 3*a^2*b^5*c*d^8 - a^
3*b^4*d^9)*x^3 + (2*b^7*c^4*d^5 - 5*a*b^6*c^3*d^6 + 3*a^2*b^5*c^2*d^7 + a^3*b^4*
c*d^8 - a^4*b^3*d^9)*x^2 + (b^7*c^5*d^4 - a*b^6*c^4*d^5 - 3*a^2*b^5*c^3*d^6 + 5*
a^3*b^4*c^2*d^7 - 2*a^4*b^3*c*d^8)*x)*sqrt(b*d)), -1/24*(2*(105*a*b^4*c^6 - 190*
a^2*b^3*c^5*d + 36*a^3*b^2*c^4*d^2 + 30*a^4*b*c^3*d^3 - 45*a^5*c^2*d^4 - 6*(b^5*
c^3*d^3 - 3*a*b^4*c^2*d^4 + 3*a^2*b^3*c*d^5 - a^3*b^2*d^6)*x^4 + 3*(7*b^5*c^4*d^
2 - 16*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 + 8*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^3 +
(140*b^5*c^5*d - 237*a*b^4*c^4*d^2 + 12*a^2*b^3*c^3*d^3 + 66*a^3*b^2*c^2*d^4 - 4
5*a^5*d^6)*x^2 + (105*b^5*c^6 - 50*a*b^4*c^5*d - 222*a^2*b^3*c^4*d^2 + 84*a^3*b^
2*c^3*d^3 + 45*a^4*b*c^2*d^4 - 90*a^5*c*d^5)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*
x + c) - 15*(7*a*b^5*c^7 - 15*a^2*b^4*c^6*d + 6*a^3*b^3*c^5*d^2 + 2*a^4*b^2*c^4*
d^3 + 3*a^5*b*c^3*d^4 - 3*a^6*c^2*d^5 + (7*b^6*c^5*d^2 - 15*a*b^5*c^4*d^3 + 6*a^
2*b^4*c^3*d^4 + 2*a^3*b^3*c^2*d^5 + 3*a^4*b^2*c*d^6 - 3*a^5*b*d^7)*x^3 + (14*b^6
*c^6*d - 23*a*b^5*c^5*d^2 - 3*a^2*b^4*c^4*d^3 + 10*a^3*b^3*c^3*d^4 + 8*a^4*b^2*c
^2*d^5 - 3*a^5*b*c*d^6 - 3*a^6*d^7)*x^2 + (7*b^6*c^7 - a*b^5*c^6*d - 24*a^2*b^4*
c^5*d^2 + 14*a^3*b^3*c^4*d^3 + 7*a^4*b^2*c^3*d^4 + 3*a^5*b*c^2*d^5 - 6*a^6*c*d^6
)*x)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*
d)))/((a*b^6*c^5*d^4 - 3*a^2*b^5*c^4*d^5 + 3*a^3*b^4*c^3*d^6 - a^4*b^3*c^2*d^7 +
(b^7*c^3*d^6 - 3*a*b^6*c^2*d^7 + 3*a^2*b^5*c*d^8 - a^3*b^4*d^9)*x^3 + (2*b^7*c^
4*d^5 - 5*a*b^6*c^3*d^6 + 3*a^2*b^5*c^2*d^7 + a^3*b^4*c*d^8 - a^4*b^3*d^9)*x^2 +
(b^7*c^5*d^4 - a*b^6*c^4*d^5 - 3*a^2*b^5*c^3*d^6 + 5*a^3*b^4*c^2*d^7 - 2*a^4*b^
3*c*d^8)*x)*sqrt(-b*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

Integral(x**5/((a + b*x)**(3/2)*(c + d*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.585965, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

sage0*x