3.727 \(\int \frac{1}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=137 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{4 a^2 c^2 x}-\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{2 a c x^2} \]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(2*a*c*x^2) + (3*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(4*a^2*c^2*x) - ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(5/2)*c^(5/2))

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Rubi [A]  time = 0.286579, antiderivative size = 137, 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 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{4 a^2 c^2 x}-\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{2 a c x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(2*a*c*x^2) + (3*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(4*a^2*c^2*x) - ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(5/2)*c^(5/2))

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Rubi in Sympy [A]  time = 29.8557, size = 117, normalized size = 0.85 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{2 a c x^{2}} + \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right )}{4 a^{2} c^{2} x} + \frac{\left (a b c d - \frac{3 \left (a d + b c\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{5}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)*sqrt(c + d*x)/(2*a*c*x**2) + 3*sqrt(a + b*x)*sqrt(c + d*x)*(a*d +
b*c)/(4*a**2*c**2*x) + (a*b*c*d - 3*(a*d + b*c)**2/4)*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 = 0.186605, size = 164, normalized size = 1.2 \[ \frac{x^2 \log (x) \left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right )-x^2 \left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (-2 a c+3 a d x+3 b c x)}{8 a^{5/2} c^{5/2} x^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-2*a*c + 3*b*c*x + 3*a*d*x) + (3
*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*x^2*Log[x] - (3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^
2)*x^2*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]
])/(8*a^(5/2)*c^(5/2)*x^2)

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Maple [B]  time = 0.037, size = 258, normalized size = 1.9 \[ -{\frac{1}{8\,{a}^{2}{c}^{2}{x}^{2}} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abcd+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{2}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dax\sqrt{ac}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }bcx\sqrt{ac}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ca\sqrt{ac} \right ) \sqrt{dx+c}\sqrt{bx+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/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^2*a^2*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)*x^
2*a*b*c*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*
b^2*c^2-6*((b*x+a)*(d*x+c))^(1/2)*d*a*x*(a*c)^(1/2)-6*((b*x+a)*(d*x+c))^(1/2)*b*
c*x*(a*c)^(1/2)+4*((b*x+a)*(d*x+c))^(1/2)*c*a*(a*c)^(1/2))*(d*x+c)^(1/2)*(b*x+a)
^(1/2)/x^2/(a*c)^(1/2)/((b*x+a)*(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/(sqrt(b*x + a)*sqrt(d*x + c)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.311592, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (2 \, a c - 3 \,{\left (b c + a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{16 \, \sqrt{a c} a^{2} c^{2} x^{2}}, -\frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (2 \, a c - 3 \,{\left (b c + a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{8 \, \sqrt{-a c} a^{2} c^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*x^2*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) - 4*(2*a*c - 3*(b*c + a*d)*
x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^2*c^2*x^2), -1/8*((3*b^2*
c^2 + 2*a*b*c*d + 3*a^2*d^2)*x^2*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(
sqrt(b*x + a)*sqrt(d*x + c)*a*c)) + 2*(2*a*c - 3*(b*c + a*d)*x)*sqrt(-a*c)*sqrt(
b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^2*c^2*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*sqrt(a + b*x)*sqrt(c + d*x)), 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/(sqrt(b*x + a)*sqrt(d*x + c)*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError