3.632 \(\int \frac{(a+b x)^{3/2}}{x (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 a \sqrt{a+b x}}{c^2 \sqrt{c+d x}}+\frac{2 (a+b x)^{3/2}}{3 c (c+d x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.169572, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 a \sqrt{a+b x}}{c^2 \sqrt{c+d x}}+\frac{2 (a+b x)^{3/2}}{3 c (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/(x*(c + d*x)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.6183, size = 85, normalized size = 0.92 \[ - \frac{2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{5}{2}}} + \frac{2 a \sqrt{a + b x}}{c^{2} \sqrt{c + d x}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 c \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)/x/(d*x+c)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.242299, size = 109, normalized size = 1.18 \[ \frac{-3 a^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+3 a^{3/2} \log (x)+\frac{2 \sqrt{c} \sqrt{a+b x} (4 a c+3 a d x+b c x)}{(c+d x)^{3/2}}}{3 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)/(x*(c + d*x)^(5/2)),x]

[Out]

((2*Sqrt[c]*Sqrt[a + b*x]*(4*a*c + b*c*x + 3*a*d*x))/(c + d*x)^(3/2) + 3*a^(3/2)
*Log[x] - 3*a^(3/2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*
Sqrt[c + d*x]])/(3*c^(5/2))

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Maple [B]  time = 0.036, size = 248, normalized size = 2.7 \[ -{\frac{1}{3\,{c}^{2}}\sqrt{bx+a} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}+6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}cd+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}{c}^{2}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dax\sqrt{ac}-2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }bcx\sqrt{ac}-8\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ca\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/x/(d*x+c)^(5/2),x)

[Out]

-1/3*(b*x+a)^(1/2)/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^2+6*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+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)*a^2*c^2-6*((b*x+a)*(d*x+c))^(1/2)*d*a*x*(a*c)^(1/2)-2*((b*x+a)*(d*x+c))^(1/2
)*b*c*x*(a*c)^(1/2)-8*((b*x+a)*(d*x+c))^(1/2)*c*a*(a*c)^(1/2))/(a*c)^(1/2)/((b*x
+a)*(d*x+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)^(3/2)/((d*x + c)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.369159, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a d^{2} x^{2} + 2 \, a c d x + a c^{2}\right )} \sqrt{\frac{a}{c}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (4 \, a c +{\left (b c + 3 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (c^{2} d^{2} x^{2} + 2 \, c^{3} d x + c^{4}\right )}}, -\frac{3 \,{\left (a d^{2} x^{2} + 2 \, a c d x + a c^{2}\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} c \sqrt{-\frac{a}{c}}}\right ) - 2 \,{\left (4 \, a c +{\left (b c + 3 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (c^{2} d^{2} x^{2} + 2 \, c^{3} d x + c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/((d*x + c)^(5/2)*x),x, algorithm="fricas")

[Out]

[1/6*(3*(a*d^2*x^2 + 2*a*c*d*x + a*c^2)*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*(4*a*c + (b*c + 3*a*d)*x)*sqr
t(b*x + a)*sqrt(d*x + c))/(c^2*d^2*x^2 + 2*c^3*d*x + c^4), -1/3*(3*(a*d^2*x^2 +
2*a*c*d*x + a*c^2)*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*(4*a*c + (b*c + 3*a*d)*x)*sqrt(b*x + a)*sqrt(d*
x + c))/(c^2*d^2*x^2 + 2*c^3*d*x + c^4)]

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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/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.267872, size = 379, normalized size = 4.12 \[ -\frac{2 \, \sqrt{b d} a^{2} b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c^{2}{\left | b \right |}} - \frac{\sqrt{b x + a}{\left (\frac{{\left (b^{5} c^{4} d{\left | b \right |} + 2 \, a b^{4} c^{3} d^{2}{\left | b \right |} - 3 \, a^{2} b^{3} c^{2} d^{3}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (a b^{5} c^{4} d{\left | b \right |} - 2 \, a^{2} b^{4} c^{3} d^{2}{\left | b \right |} + a^{3} b^{3} c^{2} d^{3}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{48 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/((d*x + c)^(5/2)*x),x, algorithm="giac")

[Out]

-2*sqrt(b*d)*a^2*b*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(
b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*c^2*abs(b
)) - 1/48*sqrt(b*x + a)*((b^5*c^4*d*abs(b) + 2*a*b^4*c^3*d^2*abs(b) - 3*a^2*b^3*
c^2*d^3*abs(b))*(b*x + a)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6) + 3*(a*b^5
*c^4*d*abs(b) - 2*a^2*b^4*c^3*d^2*abs(b) + a^3*b^3*c^2*d^3*abs(b))/(b^8*c^2*d^4
- 2*a*b^7*c*d^5 + a^2*b^6*d^6))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2)