3.787 \(\int \frac{(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{5 c^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{7/2}}-\frac{5 c \sqrt{c+d x} (b c-a d)}{a^3 \sqrt{a+b x}}-\frac{5 (c+d x)^{3/2} (b c-a d)}{3 a^2 (a+b x)^{3/2}}-\frac{(c+d x)^{5/2}}{a x (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.264275, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{5 c^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{7/2}}-\frac{5 c \sqrt{c+d x} (b c-a d)}{a^3 \sqrt{a+b x}}-\frac{5 (c+d x)^{3/2} (b c-a d)}{3 a^2 (a+b x)^{3/2}}-\frac{(c+d x)^{5/2}}{a x (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^2*(a + b*x)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 20.6427, size = 128, normalized size = 0.9 \[ \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{3 a x \left (a + b x\right )^{\frac{3}{2}}} - \frac{5 c \left (c + d x\right )^{\frac{3}{2}}}{3 a^{2} x \sqrt{a + b x}} + \frac{5 c \sqrt{c + d x} \left (a d - b c\right )}{a^{3} \sqrt{a + b x}} - \frac{5 c^{\frac{3}{2}} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**2/(b*x+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.459511, size = 162, normalized size = 1.14 \[ \frac{\frac{2 \sqrt{a} \sqrt{c+d x} \left (a^2 \left (-3 c^2+14 c d x+2 d^2 x^2\right )+10 a b c x (d x-2 c)-15 b^2 c^2 x^2\right )}{x (a+b x)^{3/2}}+15 c^{3/2} \log (x) (a d-b c)+15 c^{3/2} (b c-a d) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{6 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^2*(a + b*x)^(5/2)),x]

[Out]

((2*Sqrt[a]*Sqrt[c + d*x]*(-15*b^2*c^2*x^2 + 10*a*b*c*x*(-2*c + d*x) + a^2*(-3*c
^2 + 14*c*d*x + 2*d^2*x^2)))/(x*(a + b*x)^(3/2)) + 15*c^(3/2)*(-(b*c) + a*d)*Log
[x] + 15*c^(3/2)*(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]])/(6*a^(7/2))

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Maple [B]  time = 0.038, size = 502, normalized size = 3.5 \[ -{\frac{1}{6\,{a}^{3}x}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}b{c}^{2}d-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{3}+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{3}{c}^{2}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}b{c}^{3}-4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}-20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}+30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}-28\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}+40\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}+6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^2/(b*x+a)^(5/2),x)

[Out]

-1/6*(d*x+c)^(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*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^3*b^3*c^3+30*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^2*d-30*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^3+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*a^3*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*a^2*b*c^3-4*((b*x+a)*(d*x+c))^(1/2)*d^2*a^2*x^2*(a*c)^(1/2)-20*((b
*x+a)*(d*x+c))^(1/2)*d*b*c*a*x^2*(a*c)^(1/2)+30*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*
x^2*(a*c)^(1/2)-28*((b*x+a)*(d*x+c))^(1/2)*d*c*a^2*x*(a*c)^(1/2)+40*((b*x+a)*(d*
x+c))^(1/2)*b*c^2*a*x*(a*c)^(1/2)+6*((b*x+a)*(d*x+c))^(1/2)*c^2*a^2*(a*c)^(1/2))
/a^3/((b*x+a)*(d*x+c))^(1/2)/x/(a*c)^(1/2)/(b*x+a)^(3/2)

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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)^(5/2)/((b*x + a)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.5378, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \,{\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} +{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{\frac{c}{a}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{2} c +{\left (a b c + a^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{c}{a}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (3 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{12 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}, \frac{15 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \,{\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} +{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{-\frac{c}{a}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} a \sqrt{-\frac{c}{a}}}\right ) - 2 \,{\left (3 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^2),x, algorithm="fricas")

[Out]

[-1/12*(15*((b^3*c^2 - a*b^2*c*d)*x^3 + 2*(a*b^2*c^2 - a^2*b*c*d)*x^2 + (a^2*b*c
^2 - a^3*c*d)*x)*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2
- 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b
*c^2 + a^2*c*d)*x)/x^2) + 4*(3*a^2*c^2 + (15*b^2*c^2 - 10*a*b*c*d - 2*a^2*d^2)*x
^2 + 2*(10*a*b*c^2 - 7*a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*b^2*x^3 + 2
*a^4*b*x^2 + a^5*x), 1/6*(15*((b^3*c^2 - a*b^2*c*d)*x^3 + 2*(a*b^2*c^2 - a^2*b*c
*d)*x^2 + (a^2*b*c^2 - a^3*c*d)*x)*sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)
/(sqrt(b*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))) - 2*(3*a^2*c^2 + (15*b^2*c^2 - 10*a
*b*c*d - 2*a^2*d^2)*x^2 + 2*(10*a*b*c^2 - 7*a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x +
c))/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x)]

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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)**(5/2)/x**2/(b*x+a)**(5/2),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)^(5/2)/((b*x + a)^(5/2)*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError