3.102 \(\int \frac{x^{13/2}}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=164 \[ \frac{512 b^5 \sqrt{x}}{63 c^6 \sqrt{b x+c x^2}}+\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}} \]

[Out]

(512*b^5*Sqrt[x])/(63*c^6*Sqrt[b*x + c*x^2]) + (256*b^4*x^(3/2))/(63*c^5*Sqrt[b*
x + c*x^2]) - (64*b^3*x^(5/2))/(63*c^4*Sqrt[b*x + c*x^2]) + (32*b^2*x^(7/2))/(63
*c^3*Sqrt[b*x + c*x^2]) - (20*b*x^(9/2))/(63*c^2*Sqrt[b*x + c*x^2]) + (2*x^(11/2
))/(9*c*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.217778, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{512 b^5 \sqrt{x}}{63 c^6 \sqrt{b x+c x^2}}+\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(13/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(512*b^5*Sqrt[x])/(63*c^6*Sqrt[b*x + c*x^2]) + (256*b^4*x^(3/2))/(63*c^5*Sqrt[b*
x + c*x^2]) - (64*b^3*x^(5/2))/(63*c^4*Sqrt[b*x + c*x^2]) + (32*b^2*x^(7/2))/(63
*c^3*Sqrt[b*x + c*x^2]) - (20*b*x^(9/2))/(63*c^2*Sqrt[b*x + c*x^2]) + (2*x^(11/2
))/(9*c*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 23.9733, size = 155, normalized size = 0.95 \[ \frac{512 b^{5} \sqrt{x}}{63 c^{6} \sqrt{b x + c x^{2}}} + \frac{256 b^{4} x^{\frac{3}{2}}}{63 c^{5} \sqrt{b x + c x^{2}}} - \frac{64 b^{3} x^{\frac{5}{2}}}{63 c^{4} \sqrt{b x + c x^{2}}} + \frac{32 b^{2} x^{\frac{7}{2}}}{63 c^{3} \sqrt{b x + c x^{2}}} - \frac{20 b x^{\frac{9}{2}}}{63 c^{2} \sqrt{b x + c x^{2}}} + \frac{2 x^{\frac{11}{2}}}{9 c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(13/2)/(c*x**2+b*x)**(3/2),x)

[Out]

512*b**5*sqrt(x)/(63*c**6*sqrt(b*x + c*x**2)) + 256*b**4*x**(3/2)/(63*c**5*sqrt(
b*x + c*x**2)) - 64*b**3*x**(5/2)/(63*c**4*sqrt(b*x + c*x**2)) + 32*b**2*x**(7/2
)/(63*c**3*sqrt(b*x + c*x**2)) - 20*b*x**(9/2)/(63*c**2*sqrt(b*x + c*x**2)) + 2*
x**(11/2)/(9*c*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.0404007, size = 75, normalized size = 0.46 \[ \frac{2 \sqrt{x} \left (256 b^5+128 b^4 c x-32 b^3 c^2 x^2+16 b^2 c^3 x^3-10 b c^4 x^4+7 c^5 x^5\right )}{63 c^6 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(13/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(2*Sqrt[x]*(256*b^5 + 128*b^4*c*x - 32*b^3*c^2*x^2 + 16*b^2*c^3*x^3 - 10*b*c^4*x
^4 + 7*c^5*x^5))/(63*c^6*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.008, size = 77, normalized size = 0.5 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 7\,{x}^{5}{c}^{5}-10\,b{x}^{4}{c}^{4}+16\,{b}^{2}{x}^{3}{c}^{3}-32\,{b}^{3}{x}^{2}{c}^{2}+128\,{b}^{4}xc+256\,{b}^{5} \right ) }{63\,{c}^{6}}{x}^{{\frac{3}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(13/2)/(c*x^2+b*x)^(3/2),x)

[Out]

2/63*(c*x+b)*(7*c^5*x^5-10*b*c^4*x^4+16*b^2*c^3*x^3-32*b^3*c^2*x^2+128*b^4*c*x+2
56*b^5)*x^(3/2)/c^6/(c*x^2+b*x)^(3/2)

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Maxima [A]  time = 0.741437, size = 393, normalized size = 2.4 \[ \frac{2 \,{\left ({\left (35 \, c^{6} x^{5} - 5 \, b c^{5} x^{4} + 8 \, b^{2} c^{4} x^{3} - 16 \, b^{3} c^{3} x^{2} + 64 \, b^{4} c^{2} x + 128 \, b^{5} c\right )} x^{5} - 2 \,{\left (5 \, b c^{5} x^{5} - 2 \, b^{2} c^{4} x^{4} + 5 \, b^{3} c^{3} x^{3} - 28 \, b^{4} c^{2} x^{2} - 104 \, b^{5} c x - 64 \, b^{6}\right )} x^{4} + 6 \,{\left (3 \, b^{2} c^{4} x^{5} - 2 \, b^{3} c^{3} x^{4} + 11 \, b^{4} c^{2} x^{3} + 40 \, b^{5} c x^{2} + 24 \, b^{6} x\right )} x^{3} - 42 \,{\left (b^{3} c^{3} x^{5} - 2 \, b^{4} c^{2} x^{4} - 7 \, b^{5} c x^{3} - 4 \, b^{6} x^{2}\right )} x^{2} + 210 \,{\left (b^{4} c^{2} x^{5} + 2 \, b^{5} c x^{4} + b^{6} x^{3}\right )} x\right )}}{315 \,{\left (c^{7} x^{5} + b c^{6} x^{4}\right )} \sqrt{c x + b}} + \frac{4 \, b^{5}}{\sqrt{c x + b} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

2/315*((35*c^6*x^5 - 5*b*c^5*x^4 + 8*b^2*c^4*x^3 - 16*b^3*c^3*x^2 + 64*b^4*c^2*x
+ 128*b^5*c)*x^5 - 2*(5*b*c^5*x^5 - 2*b^2*c^4*x^4 + 5*b^3*c^3*x^3 - 28*b^4*c^2*
x^2 - 104*b^5*c*x - 64*b^6)*x^4 + 6*(3*b^2*c^4*x^5 - 2*b^3*c^3*x^4 + 11*b^4*c^2*
x^3 + 40*b^5*c*x^2 + 24*b^6*x)*x^3 - 42*(b^3*c^3*x^5 - 2*b^4*c^2*x^4 - 7*b^5*c*x
^3 - 4*b^6*x^2)*x^2 + 210*(b^4*c^2*x^5 + 2*b^5*c*x^4 + b^6*x^3)*x)/((c^7*x^5 + b
*c^6*x^4)*sqrt(c*x + b)) + 4*b^5/(sqrt(c*x + b)*c^6)

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Fricas [A]  time = 0.219962, size = 100, normalized size = 0.61 \[ \frac{2 \,{\left (7 \, c^{5} x^{6} - 10 \, b c^{4} x^{5} + 16 \, b^{2} c^{3} x^{4} - 32 \, b^{3} c^{2} x^{3} + 128 \, b^{4} c x^{2} + 256 \, b^{5} x\right )}}{63 \, \sqrt{c x^{2} + b x} c^{6} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

2/63*(7*c^5*x^6 - 10*b*c^4*x^5 + 16*b^2*c^3*x^4 - 32*b^3*c^2*x^3 + 128*b^4*c*x^2
+ 256*b^5*x)/(sqrt(c*x^2 + b*x)*c^6*sqrt(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(x**(13/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213675, size = 111, normalized size = 0.68 \[ -\frac{512 \, b^{\frac{9}{2}}}{63 \, c^{6}} + \frac{2 \,{\left (7 \,{\left (c x + b\right )}^{\frac{9}{2}} - 45 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 126 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 210 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3} + 315 \, \sqrt{c x + b} b^{4} + \frac{63 \, b^{5}}{\sqrt{c x + b}}\right )}}{63 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

-512/63*b^(9/2)/c^6 + 2/63*(7*(c*x + b)^(9/2) - 45*(c*x + b)^(7/2)*b + 126*(c*x
+ b)^(5/2)*b^2 - 210*(c*x + b)^(3/2)*b^3 + 315*sqrt(c*x + b)*b^4 + 63*b^5/sqrt(c
*x + b))/c^6