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

Optimal. Leaf size=130 \[ -\frac{2 \sqrt{a+b x} \left (a^2 d^2+b^2 c^2\right )}{b^2 d \sqrt{c+d x} (b c-a d)^2}-\frac{2 a^2}{b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.24691, antiderivative size = 130, 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 \[ -\frac{2 \sqrt{a+b x} \left (a^2 d^2+b^2 c^2\right )}{b^2 d \sqrt{c+d x} (b c-a d)^2}-\frac{2 a^2}{b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.9533, size = 117, normalized size = 0.9 \[ - \frac{2 c^{2}}{d^{2} \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )} - \frac{2 \sqrt{c + d x} \left (a^{2} d^{2} + b^{2} c^{2}\right )}{b d^{2} \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c**2/(d**2*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)) - 2*sqrt(c + d*x)*(a**2*d
**2 + b**2*c**2)/(b*d**2*sqrt(a + b*x)*(a*d - b*c)**2) + 2*atanh(sqrt(d)*sqrt(a
+ b*x)/(sqrt(b)*sqrt(c + d*x)))/(b**(3/2)*d**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.163885, size = 116, normalized size = 0.89 \[ \frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2} d^{3/2}}-\frac{2 \left (a^2 d (c+d x)+a b c^2+b^2 c^2 x\right )}{b d \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 654, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(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*d^3-2*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^2*c*d^2+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^3*c^2*d+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*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^2*b*c*d^2-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^2*c^2*d+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^3*c^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*c*d^2-2*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*c^2
*d+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^2*c^3-2*x*a^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2*x*b^2*c^2*((b*x+a)*(d
*x+c))^(1/2)*(b*d)^(1/2)-2*a^2*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2*a*b*c^2
*((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/d

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.355497, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (a b c^{2} + a^{2} c d +{\left (b^{2} c^{2} + a^{2} d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} -{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{2 \,{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} +{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} +{\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x\right )} \sqrt{b d}}, -\frac{2 \,{\left (a b c^{2} + a^{2} c d +{\left (b^{2} c^{2} + a^{2} d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} -{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} +{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} +{\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(4*(a*b*c^2 + a^2*c*d + (b^2*c^2 + a^2*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqr
t(d*x + c) - (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2
+ a^2*b*d^3)*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*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^3*c^3*d
- 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)
*x^2 + (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)*sqrt(b*d)), -(
2*(a*b*c^2 + a^2*c*d + (b^2*c^2 + a^2*d^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x
+ c) - (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2
*b*d^3)*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*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^3*c^3*d -
2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*x
^2 + (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)*sqrt(-b*d))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.596132, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x