3.120 \(\int \frac{x^4 (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=170 \[ -\frac{5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{9/2}}+\frac{5 b \sqrt{b x+c x^2} (7 b B-6 A c)}{8 c^4}-\frac{5 x \sqrt{b x+c x^2} (7 b B-6 A c)}{12 c^3}+\frac{x^2 \sqrt{b x+c x^2} (7 b B-6 A c)}{3 b c^2}-\frac{2 x^4 (b B-A c)}{b c \sqrt{b x+c x^2}} \]

[Out]

(-2*(b*B - A*c)*x^4)/(b*c*Sqrt[b*x + c*x^2]) + (5*b*(7*b*B - 6*A*c)*Sqrt[b*x + c
*x^2])/(8*c^4) - (5*(7*b*B - 6*A*c)*x*Sqrt[b*x + c*x^2])/(12*c^3) + ((7*b*B - 6*
A*c)*x^2*Sqrt[b*x + c*x^2])/(3*b*c^2) - (5*b^2*(7*b*B - 6*A*c)*ArcTanh[(Sqrt[c]*
x)/Sqrt[b*x + c*x^2]])/(8*c^(9/2))

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Rubi [A]  time = 0.378901, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{9/2}}+\frac{5 b \sqrt{b x+c x^2} (7 b B-6 A c)}{8 c^4}-\frac{5 x \sqrt{b x+c x^2} (7 b B-6 A c)}{12 c^3}+\frac{x^2 \sqrt{b x+c x^2} (7 b B-6 A c)}{3 b c^2}-\frac{2 x^4 (b B-A c)}{b c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(A + B*x))/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(b*B - A*c)*x^4)/(b*c*Sqrt[b*x + c*x^2]) + (5*b*(7*b*B - 6*A*c)*Sqrt[b*x + c
*x^2])/(8*c^4) - (5*(7*b*B - 6*A*c)*x*Sqrt[b*x + c*x^2])/(12*c^3) + ((7*b*B - 6*
A*c)*x^2*Sqrt[b*x + c*x^2])/(3*b*c^2) - (5*b^2*(7*b*B - 6*A*c)*ArcTanh[(Sqrt[c]*
x)/Sqrt[b*x + c*x^2]])/(8*c^(9/2))

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Rubi in Sympy [A]  time = 25.2197, size = 162, normalized size = 0.95 \[ \frac{5 b^{2} \left (6 A c - 7 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{9}{2}}} - \frac{5 b \left (6 A c - 7 B b\right ) \sqrt{b x + c x^{2}}}{8 c^{4}} + \frac{5 x \left (6 A c - 7 B b\right ) \sqrt{b x + c x^{2}}}{12 c^{3}} + \frac{2 x^{4} \left (A c - B b\right )}{b c \sqrt{b x + c x^{2}}} - \frac{x^{2} \left (6 A c - 7 B b\right ) \sqrt{b x + c x^{2}}}{3 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x+A)/(c*x**2+b*x)**(3/2),x)

[Out]

5*b**2*(6*A*c - 7*B*b)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(8*c**(9/2)) - 5*b*(6
*A*c - 7*B*b)*sqrt(b*x + c*x**2)/(8*c**4) + 5*x*(6*A*c - 7*B*b)*sqrt(b*x + c*x**
2)/(12*c**3) + 2*x**4*(A*c - B*b)/(b*c*sqrt(b*x + c*x**2)) - x**2*(6*A*c - 7*B*b
)*sqrt(b*x + c*x**2)/(3*b*c**2)

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Mathematica [A]  time = 0.209109, size = 132, normalized size = 0.78 \[ \frac{\sqrt{c} x \left (b^2 (35 B c x-90 A c)-2 b c^2 x (15 A+7 B x)+4 c^3 x^2 (3 A+2 B x)+105 b^3 B\right )-15 b^2 \sqrt{x} \sqrt{b+c x} (7 b B-6 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{24 c^{9/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(A + B*x))/(b*x + c*x^2)^(3/2),x]

[Out]

(Sqrt[c]*x*(105*b^3*B + 4*c^3*x^2*(3*A + 2*B*x) - 2*b*c^2*x*(15*A + 7*B*x) + b^2
*(-90*A*c + 35*B*c*x)) - 15*b^2*(7*b*B - 6*A*c)*Sqrt[x]*Sqrt[b + c*x]*Log[c*Sqrt
[x] + Sqrt[c]*Sqrt[b + c*x]])/(24*c^(9/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.012, size = 215, normalized size = 1.3 \[{\frac{A{x}^{3}}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{5\,Ab{x}^{2}}{4\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{15\,Ax{b}^{2}}{4\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{15\,{b}^{2}A}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{{x}^{4}B}{3\,c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{7\,Bb{x}^{3}}{12\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{35\,{b}^{2}B{x}^{2}}{24\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{35\,Bx{b}^{3}}{8\,{c}^{4}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{35\,B{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x+A)/(c*x^2+b*x)^(3/2),x)

[Out]

1/2*A*x^3/c/(c*x^2+b*x)^(1/2)-5/4*A*b/c^2*x^2/(c*x^2+b*x)^(1/2)-15/4*A*b^2/c^3/(
c*x^2+b*x)^(1/2)*x+15/8*A*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+
1/3*B*x^4/c/(c*x^2+b*x)^(1/2)-7/12*B*b/c^2*x^3/(c*x^2+b*x)^(1/2)+35/24*B*b^2/c^3
*x^2/(c*x^2+b*x)^(1/2)+35/8*B*b^3/c^4/(c*x^2+b*x)^(1/2)*x-35/16*B*b^3/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^4/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300072, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (8 \, B c^{3} x^{4} - 2 \,{\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{3} + 5 \,{\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{2} + 15 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x\right )} \sqrt{c}}{48 \, \sqrt{c x^{2} + b x} c^{\frac{9}{2}}}, -\frac{15 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (8 \, B c^{3} x^{4} - 2 \,{\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{3} + 5 \,{\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{2} + 15 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x\right )} \sqrt{-c}}{24 \, \sqrt{c x^{2} + b x} \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^4/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

[-1/48*(15*(7*B*b^3 - 6*A*b^2*c)*sqrt(c*x^2 + b*x)*log((2*c*x + b)*sqrt(c) + 2*s
qrt(c*x^2 + b*x)*c) - 2*(8*B*c^3*x^4 - 2*(7*B*b*c^2 - 6*A*c^3)*x^3 + 5*(7*B*b^2*
c - 6*A*b*c^2)*x^2 + 15*(7*B*b^3 - 6*A*b^2*c)*x)*sqrt(c))/(sqrt(c*x^2 + b*x)*c^(
9/2)), -1/24*(15*(7*B*b^3 - 6*A*b^2*c)*sqrt(c*x^2 + b*x)*arctan(sqrt(c*x^2 + b*x
)*sqrt(-c)/(c*x)) - (8*B*c^3*x^4 - 2*(7*B*b*c^2 - 6*A*c^3)*x^3 + 5*(7*B*b^2*c -
6*A*b*c^2)*x^2 + 15*(7*B*b^3 - 6*A*b^2*c)*x)*sqrt(-c))/(sqrt(c*x^2 + b*x)*sqrt(-
c)*c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(x**4*(A + B*x)/(x*(b + c*x))**(3/2), 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^4/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError