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

Optimal. Leaf size=175 \[ -\frac{3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} c^{7/2}}+\frac{3 \sqrt{a+b x} (b c-5 a d) (b c-a d)}{4 a c^3 \sqrt{c+d x}}-\frac{(a+b x)^{3/2} (b c-5 a d)}{4 a c^2 x \sqrt{c+d x}}-\frac{(a+b x)^{5/2}}{2 a c x^2 \sqrt{c+d x}} \]

[Out]

(3*(b*c - 5*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(4*a*c^3*Sqrt[c + d*x]) - ((b*c - 5*
a*d)*(a + b*x)^(3/2))/(4*a*c^2*x*Sqrt[c + d*x]) - (a + b*x)^(5/2)/(2*a*c*x^2*Sqr
t[c + d*x]) - (3*(b*c - 5*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt
[a]*Sqrt[c + d*x])])/(4*Sqrt[a]*c^(7/2))

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Rubi [A]  time = 0.309918, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} c^{7/2}}+\frac{3 \sqrt{a+b x} (b c-5 a d) (b c-a d)}{4 a c^3 \sqrt{c+d x}}-\frac{(a+b x)^{3/2} (b c-5 a d)}{4 a c^2 x \sqrt{c+d x}}-\frac{(a+b x)^{5/2}}{2 a c x^2 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*c - 5*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(4*a*c^3*Sqrt[c + d*x]) - ((b*c - 5*
a*d)*(a + b*x)^(3/2))/(4*a*c^2*x*Sqrt[c + d*x]) - (a + b*x)^(5/2)/(2*a*c*x^2*Sqr
t[c + d*x]) - (3*(b*c - 5*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt
[a]*Sqrt[c + d*x])])/(4*Sqrt[a]*c^(7/2))

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Rubi in Sympy [A]  time = 27.1813, size = 163, normalized size = 0.93 \[ \frac{2 d \left (a + b x\right )^{\frac{5}{2}}}{c x^{2} \sqrt{c + d x} \left (a d - b c\right )} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (5 a d - b c\right )}{2 c^{2} x^{2} \left (a d - b c\right )} + \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (5 a d - b c\right )}{4 c^{3} x} - \frac{3 \left (a d - b c\right ) \left (5 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 \sqrt{a} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d*(a + b*x)**(5/2)/(c*x**2*sqrt(c + d*x)*(a*d - b*c)) - (a + b*x)**(3/2)*sqrt(
c + d*x)*(5*a*d - b*c)/(2*c**2*x**2*(a*d - b*c)) + 3*sqrt(a + b*x)*sqrt(c + d*x)
*(5*a*d - b*c)/(4*c**3*x) - 3*(a*d - b*c)*(5*a*d - b*c)*atanh(sqrt(c)*sqrt(a + b
*x)/(sqrt(a)*sqrt(c + d*x)))/(4*sqrt(a)*c**(7/2))

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Mathematica [A]  time = 0.212686, size = 165, normalized size = 0.94 \[ \frac{\frac{2 \sqrt{c} \sqrt{a+b x} \left (a \left (-2 c^2+5 c d x+15 d^2 x^2\right )-b c x (5 c+13 d x)\right )}{x^2 \sqrt{c+d x}}+\frac{3 \log (x) (b c-5 a d) (b c-a d)}{\sqrt{a}}-\frac{3 (b c-5 a d) (b c-a d) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}}{8 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*Sqrt[a + b*x]*(-(b*c*x*(5*c + 13*d*x)) + a*(-2*c^2 + 5*c*d*x + 15*d^
2*x^2)))/(x^2*Sqrt[c + d*x]) + (3*(b*c - 5*a*d)*(b*c - a*d)*Log[x])/Sqrt[a] - (3
*(b*c - 5*a*d)*(b*c - 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]])/Sqrt[a])/(8*c^(7/2))

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Maple [B]  time = 0.038, size = 464, normalized size = 2.7 \[ -{\frac{1}{8\,{c}^{3}{x}^{2}}\sqrt{bx+a} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}{d}^{3}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}abc{d}^{2}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{2}{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}c{d}^{2}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}ab{c}^{2}d+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{3}-30\,{x}^{2}a{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+26\,{x}^{2}bcd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-10\,xacd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+10\,xb{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+4\,a{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(b*x+a)^(1/2)*(15*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*d^3-18*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*c*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^2*c^2*d+15*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*c*d^2-18*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^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^2*b^2*c^3-30*x^2*a*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+26*x^2*b*c*
d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-10*x*a*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(
1/2)+10*x*b*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*a*c^2*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2))/c^3/((b*x+a)*(d*x+c))^(1/2)/x^2/(a*c)^(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((b*x + a)^(3/2)/((d*x + c)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.419668, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (2 \, a c^{2} +{\left (13 \, b c d - 15 \, a d^{2}\right )} x^{2} + 5 \,{\left (b c^{2} - a c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{2}\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{16 \,{\left (c^{3} d x^{3} + c^{4} x^{2}\right )} \sqrt{a c}}, -\frac{2 \,{\left (2 \, a c^{2} +{\left (13 \, b c d - 15 \, a d^{2}\right )} x^{2} + 5 \,{\left (b c^{2} - a c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{8 \,{\left (c^{3} d x^{3} + c^{4} x^{2}\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*(2*a*c^2 + (13*b*c*d - 15*a*d^2)*x^2 + 5*(b*c^2 - a*c*d)*x)*sqrt(a*c)*
sqrt(b*x + a)*sqrt(d*x + c) - 3*((b^2*c^2*d - 6*a*b*c*d^2 + 5*a^2*d^3)*x^3 + (b^
2*c^3 - 6*a*b*c^2*d + 5*a^2*c*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))/((c^3*d*x^3 + c^4*x^2)*sqrt(a*c))
, -1/8*(2*(2*a*c^2 + (13*b*c*d - 15*a*d^2)*x^2 + 5*(b*c^2 - a*c*d)*x)*sqrt(-a*c)
*sqrt(b*x + a)*sqrt(d*x + c) + 3*((b^2*c^2*d - 6*a*b*c*d^2 + 5*a^2*d^3)*x^3 + (b
^2*c^3 - 6*a*b*c^2*d + 5*a^2*c*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)))/((c^3*d*x^3 + c^4*x^2)*sqrt(-a*c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

[Out]

Exception raised: TypeError