3.129 \(\int \frac{x^5 (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=172 \[ \frac{5 b (7 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{9/2}}-\frac{5 \sqrt{b x+c x^2} (7 b B-4 A c)}{4 c^4}+\frac{5 x \sqrt{b x+c x^2} (7 b B-4 A c)}{6 b c^3}-\frac{2 x^3 (7 b B-4 A c)}{3 b c^2 \sqrt{b x+c x^2}}-\frac{2 x^5 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(b*B - A*c)*x^5)/(3*b*c*(b*x + c*x^2)^(3/2)) - (2*(7*b*B - 4*A*c)*x^3)/(3*b*
c^2*Sqrt[b*x + c*x^2]) - (5*(7*b*B - 4*A*c)*Sqrt[b*x + c*x^2])/(4*c^4) + (5*(7*b
*B - 4*A*c)*x*Sqrt[b*x + c*x^2])/(6*b*c^3) + (5*b*(7*b*B - 4*A*c)*ArcTanh[(Sqrt[
c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(9/2))

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Rubi [A]  time = 0.402652, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{5 b (7 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{9/2}}-\frac{5 \sqrt{b x+c x^2} (7 b B-4 A c)}{4 c^4}+\frac{5 x \sqrt{b x+c x^2} (7 b B-4 A c)}{6 b c^3}-\frac{2 x^3 (7 b B-4 A c)}{3 b c^2 \sqrt{b x+c x^2}}-\frac{2 x^5 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*B - A*c)*x^5)/(3*b*c*(b*x + c*x^2)^(3/2)) - (2*(7*b*B - 4*A*c)*x^3)/(3*b*
c^2*Sqrt[b*x + c*x^2]) - (5*(7*b*B - 4*A*c)*Sqrt[b*x + c*x^2])/(4*c^4) + (5*(7*b
*B - 4*A*c)*x*Sqrt[b*x + c*x^2])/(6*b*c^3) + (5*b*(7*b*B - 4*A*c)*ArcTanh[(Sqrt[
c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(9/2))

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Rubi in Sympy [A]  time = 25.3463, size = 163, normalized size = 0.95 \[ - \frac{5 b \left (4 A c - 7 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 c^{\frac{9}{2}}} + \frac{5 \left (4 A c - 7 B b\right ) \sqrt{b x + c x^{2}}}{4 c^{4}} + \frac{2 x^{5} \left (A c - B b\right )}{3 b c \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{2 x^{3} \left (4 A c - 7 B b\right )}{3 b c^{2} \sqrt{b x + c x^{2}}} - \frac{5 x \left (4 A c - 7 B b\right ) \sqrt{b x + c x^{2}}}{6 b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(B*x+A)/(c*x**2+b*x)**(5/2),x)

[Out]

-5*b*(4*A*c - 7*B*b)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(4*c**(9/2)) + 5*(4*A*c
- 7*B*b)*sqrt(b*x + c*x**2)/(4*c**4) + 2*x**5*(A*c - B*b)/(3*b*c*(b*x + c*x**2)
**(3/2)) + 2*x**3*(4*A*c - 7*B*b)/(3*b*c**2*sqrt(b*x + c*x**2)) - 5*x*(4*A*c - 7
*B*b)*sqrt(b*x + c*x**2)/(6*b*c**3)

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Mathematica [A]  time = 0.22294, size = 129, normalized size = 0.75 \[ \frac{x \left (\sqrt{c} x \left (20 b^2 c (3 A-7 B x)+b c^2 x (80 A-21 B x)+6 c^3 x^2 (2 A+B x)-105 b^3 B\right )+15 b \sqrt{x} (b+c x)^{3/2} (7 b B-4 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )}{12 c^{9/2} (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(Sqrt[c]*x*(-105*b^3*B + b*c^2*x*(80*A - 21*B*x) + 20*b^2*c*(3*A - 7*B*x) + 6
*c^3*x^2*(2*A + B*x)) + 15*b*(7*b*B - 4*A*c)*Sqrt[x]*(b + c*x)^(3/2)*Log[c*Sqrt[
x] + Sqrt[c]*Sqrt[b + c*x]]))/(12*c^(9/2)*(x*(b + c*x))^(3/2))

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Maple [B]  time = 0.016, size = 338, normalized size = 2. \[{\frac{A{x}^{4}}{c} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,Ab{x}^{3}}{6\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,A{b}^{2}{x}^{2}}{4\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,A{b}^{3}x}{12\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,Abx}{6\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{5\,{b}^{2}A}{12\,{c}^{4}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{5\,Ab}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{{x}^{5}B}{2\,c} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,Bb{x}^{4}}{4\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{b}^{2}B{x}^{3}}{24\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,B{b}^{3}{x}^{2}}{16\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{b}^{4}Bx}{48\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}-{\frac{245\,{b}^{2}Bx}{24\,{c}^{4}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{35\,B{b}^{3}}{48\,{c}^{5}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{35\,{b}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*x^4/c/(c*x^2+b*x)^(3/2)+5/6*A*b/c^2*x^3/(c*x^2+b*x)^(3/2)-5/4*A*b^2/c^3*x^2/(c
*x^2+b*x)^(3/2)-5/12*A*b^3/c^4/(c*x^2+b*x)^(3/2)*x+35/6*A*b/c^3/(c*x^2+b*x)^(1/2
)*x+5/12*A*b^2/c^4/(c*x^2+b*x)^(1/2)-5/2*A*b/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x)^(1/2))+1/2*B*x^5/c/(c*x^2+b*x)^(3/2)-7/4*B*b/c^2*x^4/(c*x^2+b*x)^(3/2)-3
5/24*B*b^2/c^3*x^3/(c*x^2+b*x)^(3/2)+35/16*B*b^3/c^4*x^2/(c*x^2+b*x)^(3/2)+35/48
*B*b^4/c^5/(c*x^2+b*x)^(3/2)*x-245/24*B*b^2/c^4/(c*x^2+b*x)^(1/2)*x-35/48*B*b^3/
c^5/(c*x^2+b*x)^(1/2)+35/8*B*b^2/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(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)*x^5/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.314208, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B b^{3} - 4 \, A b^{2} c +{\left (7 \, B b^{2} c - 4 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (6 \, B c^{3} x^{4} - 3 \,{\left (7 \, B b c^{2} - 4 \, A c^{3}\right )} x^{3} - 20 \,{\left (7 \, B b^{2} c - 4 \, A b c^{2}\right )} x^{2} - 15 \,{\left (7 \, B b^{3} - 4 \, A b^{2} c\right )} x\right )} \sqrt{c}}{24 \,{\left (c^{5} x + b c^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}, \frac{15 \,{\left (7 \, B b^{3} - 4 \, A b^{2} c +{\left (7 \, B b^{2} c - 4 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (6 \, B c^{3} x^{4} - 3 \,{\left (7 \, B b c^{2} - 4 \, A c^{3}\right )} x^{3} - 20 \,{\left (7 \, B b^{2} c - 4 \, A b c^{2}\right )} x^{2} - 15 \,{\left (7 \, B b^{3} - 4 \, A b^{2} c\right )} x\right )} \sqrt{-c}}{12 \,{\left (c^{5} x + b c^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/24*(15*(7*B*b^3 - 4*A*b^2*c + (7*B*b^2*c - 4*A*b*c^2)*x)*sqrt(c*x^2 + b*x)*l
og((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c) - 2*(6*B*c^3*x^4 - 3*(7*B*b*c^2
- 4*A*c^3)*x^3 - 20*(7*B*b^2*c - 4*A*b*c^2)*x^2 - 15*(7*B*b^3 - 4*A*b^2*c)*x)*sq
rt(c))/((c^5*x + b*c^4)*sqrt(c*x^2 + b*x)*sqrt(c)), 1/12*(15*(7*B*b^3 - 4*A*b^2*
c + (7*B*b^2*c - 4*A*b*c^2)*x)*sqrt(c*x^2 + b*x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-
c)/(c*x)) + (6*B*c^3*x^4 - 3*(7*B*b*c^2 - 4*A*c^3)*x^3 - 20*(7*B*b^2*c - 4*A*b*c
^2)*x^2 - 15*(7*B*b^3 - 4*A*b^2*c)*x)*sqrt(-c))/((c^5*x + b*c^4)*sqrt(c*x^2 + b*
x)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(B*x+A)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral(x**5*(A + B*x)/(x*(b + c*x))**(5/2), x)

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

[Out]

Exception raised: TypeError