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

Optimal. Leaf size=178 \[ -\frac{3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} \sqrt{c}}+\frac{3 \sqrt{c+d x} (b c-a d) (5 b c-a d)}{4 a^3 c \sqrt{a+b x}}+\frac{(c+d x)^{3/2} (5 b c-a d)}{4 a^2 c x \sqrt{a+b x}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.313232, antiderivative size = 178, 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 \[ -\frac{3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} \sqrt{c}}+\frac{3 \sqrt{c+d x} (b c-a d) (5 b c-a d)}{4 a^3 c \sqrt{a+b x}}+\frac{(c+d x)^{3/2} (5 b c-a d)}{4 a^2 c x \sqrt{a+b x}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.1059, size = 165, normalized size = 0.93 \[ - \frac{2 b \left (c + d x\right )^{\frac{5}{2}}}{a x^{2} \sqrt{a + b x} \left (a d - b c\right )} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - 5 b c\right )}{2 a^{2} x^{2} \left (a d - b c\right )} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - 5 b c\right )}{4 a^{3} x} - \frac{3 \left (a d - 5 b c\right ) \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{7}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.220275, size = 165, normalized size = 0.93 \[ \frac{\frac{2 \sqrt{a} \sqrt{c+d x} \left (a^2 (-(2 c+5 d x))+a b x (5 c-13 d x)+15 b^2 c x^2\right )}{x^2 \sqrt{a+b x}}+\frac{3 \log (x) (a d-5 b c) (a d-b c)}{\sqrt{c}}-\frac{3 (a d-5 b c) (a d-b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}}{8 a^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.04, size = 464, normalized size = 2.6 \[ -{\frac{1}{8\,{a}^{3}{x}^{2}}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}b{d}^{2}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}cd+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{2}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{3}{d}^{2}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}bcd+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{2}+26\,{x}^{2}abd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-30\,{x}^{2}{b}^{2}c\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+10\,x{a}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-10\,xabc\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+4\,{a}^{2}c\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.451908, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (2 \, a^{2} c -{\left (15 \, b^{2} c - 13 \, a b d\right )} x^{2} - 5 \,{\left (a b c - a^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{16 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )} \sqrt{a c}}, -\frac{2 \,{\left (2 \, a^{2} c -{\left (15 \, b^{2} c - 13 \, a b d\right )} x^{2} - 5 \,{\left (a b c - a^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{8 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError