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

Optimal. Leaf size=167 \[ \frac{\sqrt{a+b x} \left (c \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )+d x (b c-3 a d) (b c-a d)\right )}{b^2 d^2 \sqrt{c+d x} (b c-a d)^2}-\frac{3 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac{2 a x^2}{b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)} \]

[Out]

(2*a*x^2)/(b*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) + (Sqrt[a + b*x]*(c*(3*b^2
*c^2 - 2*a*b*c*d + 3*a^2*d^2) + d*(b*c - 3*a*d)*(b*c - a*d)*x))/(b^2*d^2*(b*c -
a*d)^2*Sqrt[c + d*x]) - (3*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(b^(5/2)*d^(5/2))

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Rubi [A]  time = 0.363497, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt{a+b x} \left (c \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )+d x (b c-3 a d) (b c-a d)\right )}{b^2 d^2 \sqrt{c+d x} (b c-a d)^2}-\frac{3 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac{2 a x^2}{b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*x^2)/(b*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) + (Sqrt[a + b*x]*(c*(3*b^2
*c^2 - 2*a*b*c*d + 3*a^2*d^2) + d*(b*c - 3*a*d)*(b*c - a*d)*x))/(b^2*d^2*(b*c -
a*d)^2*Sqrt[c + d*x]) - (3*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(b^(5/2)*d^(5/2))

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Rubi in Sympy [A]  time = 26.6098, size = 162, normalized size = 0.97 \[ - \frac{2 a x^{2}}{b \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )} + \frac{4 \sqrt{a + b x} \left (\frac{c \left (3 a^{2} d^{2} - 2 a b c d + 3 b^{2} c^{2}\right )}{4} + \frac{d x \left (a d - b c\right ) \left (3 a d - b c\right )}{4}\right )}{b^{2} d^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{3 \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*x**2/(b*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)) + 4*sqrt(a + b*x)*(c*(3*a*
*2*d**2 - 2*a*b*c*d + 3*b**2*c**2)/4 + d*x*(a*d - b*c)*(3*a*d - b*c)/4)/(b**2*d*
*2*sqrt(c + d*x)*(a*d - b*c)**2) - 3*(a*d + b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(sq
rt(b)*sqrt(c + d*x)))/(b**(5/2)*d**(5/2))

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Mathematica [A]  time = 0.409155, size = 142, normalized size = 0.85 \[ \sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 a^3}{b^2 (a+b x) (b c-a d)^2}+\frac{2 c^3}{d^2 (c+d x) (a d-b c)^2}+\frac{1}{b^2 d^2}\right )-\frac{3 (a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*(1/(b^2*d^2) + (2*a^3)/(b^2*(b*c - a*d)^2*(a + b*x))
+ (2*c^3)/(d^2*(-(b*c) + a*d)^2*(c + d*x))) - (3*(b*c + a*d)*Log[b*c + a*d + 2*
b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*b^(5/2)*d^(5/2))

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Maple [B]  time = 0.037, size = 906, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(3*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*d^4-3*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^2*c*d^3-3*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^3*c^2*d^2+3*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^4*c^3*d+3*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*d^4-6*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^2*c^
2*d^2+3*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^4*c^4-2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a^2*b*d^3+4*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)*x^2*a*b^2*c*d^2-2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^
2*b^3*c^2*d+3*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*c*d^3-3*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*c^2*d^2-3*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^2*c^3*d+3*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^3*c^4-6*x*a^3*d^3*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^2*b*c*d^2+2*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a*b^2*c^2*d-6*x*b^3*c^3*((b*x+a)*(d*x+c))^(1/2)*
(b*d)^(1/2)-6*a^3*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+4*((b*x+a)*(d*x+c))^
(1/2)*(b*d)^(1/2)*a^2*b*c^2*d-6*a*b^2*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(
(b*x+a)*(d*x+c))^(1/2)/(a*d-b*c)^2/(b*d)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/b^2/d
^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^3/((b*x + a)^(3/2)*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.423249, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*(3*a*b^2*c^3 - 2*a^2*b*c^2*d + 3*a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2
+ a^2*b*d^3)*x^2 + (3*b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + 3*a^3*d^3)*x)*sqrt(b
*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 +
a^4*c*d^3 + (b^4*c^3*d - a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + (b^4*
c^4 - 2*a^2*b^2*c^2*d^2 + a^4*d^4)*x)*log(-4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*s
qrt(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^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 + a^3*b^2
*c*d^4 + (b^5*c^2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*x^2 + (b^5*c^3*d^2 - a*b^4*
c^2*d^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*x)*sqrt(b*d)), 1/2*(2*(3*a*b^2*c^3 - 2*a^
2*b*c^2*d + 3*a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + (3*b^3*c
^3 - a*b^2*c^2*d - a^2*b*c*d^2 + 3*a^3*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x
+ c) - 3*(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3 + (b^4*c^3*d -
a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + (b^4*c^4 - 2*a^2*b^2*c^2*d^2 +
a^4*d^4)*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^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4 + (b^5*c^2*d^3 -
2*a*b^4*c*d^4 + a^2*b^3*d^5)*x^2 + (b^5*c^3*d^2 - a*b^4*c^2*d^3 - a^2*b^3*c*d^4
+ a^3*b^2*d^5)*x)*sqrt(-b*d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.555284, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x