3.770 \(\int \frac{1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=185 \[ \frac{3 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2} c^{5/2}}-\frac{d \sqrt{a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a^2 c^2 \sqrt{c+d x} (b c-a d)^2}-\frac{b (3 b c-a d)}{a^2 c \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}-\frac{1}{a c x \sqrt{a+b x} \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.570617, antiderivative size = 185, 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 \[ \frac{3 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2} c^{5/2}}-\frac{d \sqrt{a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{a^2 c^2 \sqrt{c+d x} (b c-a d)^2}-\frac{b (3 b c-a d)}{a^2 c \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}-\frac{1}{a c x \sqrt{a+b x} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 71.4362, size = 168, normalized size = 0.91 \[ - \frac{1}{a c x \sqrt{a + b x} \sqrt{c + d x}} - \frac{b \left (a d - 3 b c\right )}{a^{2} c \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )} - \frac{d \sqrt{a + b x} \left (3 a^{2} d^{2} - 2 a b c d + 3 b^{2} c^{2}\right )}{a^{2} c^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{3 \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{5}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

-1/(a*c*x*sqrt(a + b*x)*sqrt(c + d*x)) - b*(a*d - 3*b*c)/(a**2*c*sqrt(a + b*x)*s
qrt(c + d*x)*(a*d - b*c)) - d*sqrt(a + b*x)*(3*a**2*d**2 - 2*a*b*c*d + 3*b**2*c*
*2)/(a**2*c**2*sqrt(c + d*x)*(a*d - b*c)**2) + 3*(a*d + b*c)*atanh(sqrt(c)*sqrt(
a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(a**(5/2)*c**(5/2))

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Mathematica [A]  time = 0.672259, size = 170, normalized size = 0.92 \[ -\frac{3 \log (x) (a d+b c)}{2 a^{5/2} c^{5/2}}+\frac{3 (a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 a^{5/2} c^{5/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (-\frac{2 b^3}{a^2 (a+b x) (a d-b c)^2}-\frac{1}{a^2 c^2 x}-\frac{2 d^3}{c^2 (c+d x) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

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

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Maple [B]  time = 0.057, size = 897, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^(3/2)/(d*x+c)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.382291, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Integral(1/(x**2*(a + b*x)**(3/2)*(c + d*x)**(3/2)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError