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

Optimal. Leaf size=121 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}}+\frac{2 b}{a \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 d \sqrt{a+b x} (a d+b c)}{a c \sqrt{c+d x} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.30431, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}}+\frac{2 b}{a \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 d \sqrt{a+b x} (a d+b c)}{a c \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.4658, size = 107, normalized size = 0.88 \[ - \frac{2 b}{a \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )} + \frac{2 d \sqrt{a + b x} \left (a d + b c\right )}{a c \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.209966, size = 125, normalized size = 1.03 \[ \frac{\frac{2 \sqrt{a} \sqrt{c} \left (a^2 d^2+a b d^2 x+b^2 c (c+d x)\right )}{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\log (x)}{a^{3/2} c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.05, size = 638, normalized size = 5.3 \[{\frac{1}{ac \left ( ad-bc \right ) ^{2}} \left ( -\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{a}^{2}b{d}^{3}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}c{d}^{2}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{b}^{3}{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{3}{d}^{3}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{2}bc{d}^{2}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xa{b}^{2}{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{b}^{3}{c}^{3}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){a}^{3}c{d}^{2}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}b{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) a{b}^{2}{c}^{3}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}xab{d}^{2}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}x{b}^{2}cd+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{a}^{2}{d}^{2}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{b}^{2}{c}^{2} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.356424, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (b^{2} c^{2} + a^{2} d^{2} +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} +{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{2 \,{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} +{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{2} +{\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x\right )} \sqrt{a c}}, \frac{2 \,{\left (b^{2} c^{2} + a^{2} d^{2} +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} -{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} +{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{2} +{\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.986374, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x