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

Optimal. Leaf size=124 \[ \frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{5/2}}-\frac{d \sqrt{a+b x} (b c-3 a d)}{a c^2 \sqrt{c+d x} (b c-a d)}-\frac{\sqrt{a+b x}}{a c x \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.315748, antiderivative size = 124, 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 \[ \frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{5/2}}-\frac{d \sqrt{a+b x} (b c-3 a d)}{a c^2 \sqrt{c+d x} (b c-a d)}-\frac{\sqrt{a+b x}}{a c x \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.5837, size = 107, normalized size = 0.86 \[ - \frac{\sqrt{a + b x}}{a c x \sqrt{c + d x}} - \frac{d \sqrt{a + b x} \left (3 a d - b c\right )}{a c^{2} \sqrt{c + d x} \left (a d - b c\right )} + \frac{\left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)/(a*c*x*sqrt(c + d*x)) - d*sqrt(a + b*x)*(3*a*d - b*c)/(a*c**2*sqr
t(c + d*x)*(a*d - b*c)) + (3*a*d + b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqr
t(c + d*x)))/(a**(3/2)*c**(5/2))

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Mathematica [A]  time = 0.338947, size = 141, normalized size = 1.14 \[ \frac{-\frac{\log (x) (3 a d+b c)}{a^{3/2}}+\frac{(3 a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2}}+2 \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 d^2}{(c+d x) (b c-a d)}-\frac{1}{a x}\right )}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.046, size = 441, normalized size = 3.6 \[{\frac{1}{2\, \left ( ad-bc \right ) a{c}^{2}x}\sqrt{bx+a} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{3}-2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abc{d}^{2}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{2}d+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}c{d}^{2}-2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xab{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{b}^{2}{c}^{3}-6\,xa{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,xbcd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-2\,acd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,b{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x+a)^(1/2)/a/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^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*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^2*c^2*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*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)*x*a*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)
*x*b^2*c^3-6*x*a*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*x*b*c*d*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)-2*a*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*b*c^2*(a*c)
^(1/2)*((b*x+a)*(d*x+c))^(1/2))/(a*d-b*c)/(a*c)^(1/2)/x/((b*x+a)*(d*x+c))^(1/2)/
(d*x+c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(3/2)*x^2), x)

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Fricas [A]  time = 0.325383, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (b c^{2} - a c d +{\left (b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} -{\left ({\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} +{\left (b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{4 \,{\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{2} +{\left (a b c^{4} - a^{2} c^{3} d\right )} x\right )} \sqrt{a c}}, -\frac{2 \,{\left (b c^{2} - a c d +{\left (b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} -{\left ({\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} +{\left (b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{2 \,{\left ({\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{2} +{\left (a b c^{4} - a^{2} c^{3} d\right )} x\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*sqrt(a + b*x)*(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/(sqrt(b*x + a)*(d*x + c)^(3/2)*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError