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

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

[Out]

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

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Rubi [A]  time = 0.264281, 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 \[ -\frac{5 a^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{7/2}}+\frac{5 a \sqrt{a+b x} (b c-a d)}{c^3 \sqrt{c+d x}}+\frac{5 (a+b x)^{3/2} (b c-a d)}{3 c^2 (c+d x)^{3/2}}-\frac{(a+b x)^{5/2}}{c x (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.8416, size = 124, normalized size = 0.87 \[ \frac{5 a^{\frac{3}{2}} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{7}{2}}} - \frac{5 a^{2} \sqrt{a + b x} \sqrt{c + d x}}{c^{3} x} + \frac{10 a \left (a + b x\right )^{\frac{3}{2}}}{3 c^{2} x \sqrt{c + d x}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{3 c x \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(3/2)*(a*d - b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/c**(
7/2) - 5*a**2*sqrt(a + b*x)*sqrt(c + d*x)/(c**3*x) + 10*a*(a + b*x)**(3/2)/(3*c*
*2*x*sqrt(c + d*x)) + 2*(a + b*x)**(5/2)/(3*c*x*(c + d*x)**(3/2))

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Mathematica [A]  time = 0.466509, size = 164, normalized size = 1.15 \[ \frac{-15 a^{3/2} \log (x) (a d-b c)+15 a^{3/2} (a d-b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{2 \sqrt{c} \sqrt{a+b x} \left (a^2 \left (-\left (3 c^2+20 c d x+15 d^2 x^2\right )\right )+2 a b c x (7 c+5 d x)+2 b^2 c^2 x^2\right )}{x (c+d x)^{3/2}}}{6 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*Sqrt[a + b*x]*(2*b^2*c^2*x^2 + 2*a*b*c*x*(7*c + 5*d*x) - a^2*(3*c^2
+ 20*c*d*x + 15*d^2*x^2)))/(x*(c + d*x)^(3/2)) - 15*a^(3/2)*(-(b*c) + a*d)*Log[x
] + 15*a^(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*c^(7/2))

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Maple [B]  time = 0.039, size = 502, normalized size = 3.5 \[{\frac{1}{6\,{c}^{3}x}\sqrt{bx+a} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{3}c{d}^{2}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}b{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{3}{c}^{2}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}b{c}^{3}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}+20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}-40\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}+28\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(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^3*d^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^3*a^2*b*c*d^2+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^3*c*d^2-30*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+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^3-30*((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)+4*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*x
^2*(a*c)^(1/2)-40*((b*x+a)*(d*x+c))^(1/2)*d*c*a^2*x*(a*c)^(1/2)+28*((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))/
c^3/((b*x+a)*(d*x+c))^(1/2)/x/(a*c)^(1/2)/(d*x+c)^(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((b*x + a)^(5/2)/((d*x + c)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.489635, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 2 \,{\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2} +{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{\frac{a}{c}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (3 \, a^{2} c^{2} -{\left (2 \, b^{2} c^{2} + 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} - 2 \,{\left (7 \, a b c^{2} - 10 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{12 \,{\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}, -\frac{15 \,{\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 2 \,{\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2} +{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} c \sqrt{-\frac{a}{c}}}\right ) + 2 \,{\left (3 \, a^{2} c^{2} -{\left (2 \, b^{2} c^{2} + 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} - 2 \,{\left (7 \, a b c^{2} - 10 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError