3.305 \(\int \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=118 \[ -\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c} \]

[Out]

(5*b^4*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^3) - (5*b^2*(b + 2*c*x)*(b*x + c*x^
2)^(3/2))/(192*c^2) + ((b + 2*c*x)*(b*x + c*x^2)^(5/2))/(12*c) - (5*b^6*ArcTanh[
(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(7/2))

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Rubi [A]  time = 0.099044, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^4*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^3) - (5*b^2*(b + 2*c*x)*(b*x + c*x^
2)^(3/2))/(192*c^2) + ((b + 2*c*x)*(b*x + c*x^2)^(5/2))/(12*c) - (5*b^6*ArcTanh[
(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(7/2))

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Rubi in Sympy [A]  time = 11.8823, size = 110, normalized size = 0.93 \[ - \frac{5 b^{6} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{512 c^{\frac{7}{2}}} + \frac{5 b^{4} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{512 c^{3}} - \frac{5 b^{2} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{192 c^{2}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{12 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*b**6*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(512*c**(7/2)) + 5*b**4*(b + 2*c*x)*
sqrt(b*x + c*x**2)/(512*c**3) - 5*b**2*(b + 2*c*x)*(b*x + c*x**2)**(3/2)/(192*c*
*2) + (b + 2*c*x)*(b*x + c*x**2)**(5/2)/(12*c)

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Mathematica [A]  time = 0.139924, size = 122, normalized size = 1.03 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (15 b^5-10 b^4 c x+8 b^3 c^2 x^2+432 b^2 c^3 x^3+640 b c^4 x^4+256 c^5 x^5\right )-\frac{15 b^6 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{1536 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^5 - 10*b^4*c*x + 8*b^3*c^2*x^2 + 432*b^2*c^3*x
^3 + 640*b*c^4*x^4 + 256*c^5*x^5) - (15*b^6*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x
]])/(Sqrt[x]*Sqrt[b + c*x])))/(1536*c^(7/2))

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Maple [A]  time = 0.006, size = 134, normalized size = 1.1 \[{\frac{2\,cx+b}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}x}{96\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{3}}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{4}x}{256\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{5}}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(5/2),x)

[Out]

1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/96*b^2/c*(c*x^2+b*x)^(3/2)*x-5/192*b^3/c^2*
(c*x^2+b*x)^(3/2)+5/256*b^4/c^2*(c*x^2+b*x)^(1/2)*x+5/512*b^5/c^3*(c*x^2+b*x)^(1
/2)-5/1024*b^6/c^(7/2)*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((c*x^2 + b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231709, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{6} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (256 \, c^{5} x^{5} + 640 \, b c^{4} x^{4} + 432 \, b^{2} c^{3} x^{3} + 8 \, b^{3} c^{2} x^{2} - 10 \, b^{4} c x + 15 \, b^{5}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{3072 \, c^{\frac{7}{2}}}, -\frac{15 \, b^{6} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (256 \, c^{5} x^{5} + 640 \, b c^{4} x^{4} + 432 \, b^{2} c^{3} x^{3} + 8 \, b^{3} c^{2} x^{2} - 10 \, b^{4} c x + 15 \, b^{5}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{1536 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3072*(15*b^6*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c) + 2*(256*c^5*x^
5 + 640*b*c^4*x^4 + 432*b^2*c^3*x^3 + 8*b^3*c^2*x^2 - 10*b^4*c*x + 15*b^5)*sqrt(
c*x^2 + b*x)*sqrt(c))/c^(7/2), -1/1536*(15*b^6*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)
/(c*x)) - (256*c^5*x^5 + 640*b*c^4*x^4 + 432*b^2*c^3*x^3 + 8*b^3*c^2*x^2 - 10*b^
4*c*x + 15*b^5)*sqrt(c*x^2 + b*x)*sqrt(-c))/(sqrt(-c)*c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b*x + c*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.234505, size = 144, normalized size = 1.22 \[ \frac{5 \, b^{6}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{7}{2}}} + \frac{1}{1536} \, \sqrt{c x^{2} + b x}{\left (\frac{15 \, b^{5}}{c^{3}} - 2 \,{\left (\frac{5 \, b^{4}}{c^{2}} - 4 \,{\left (\frac{b^{3}}{c} + 2 \,{\left (27 \, b^{2} + 8 \,{\left (2 \, c^{2} x + 5 \, b c\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/1024*b^6*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(7/2) + 1/1
536*sqrt(c*x^2 + b*x)*(15*b^5/c^3 - 2*(5*b^4/c^2 - 4*(b^3/c + 2*(27*b^2 + 8*(2*c
^2*x + 5*b*c)*x)*x)*x)*x)