3.2963 \(\int x^{17} \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{4 a^3 \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^4 c^6}+\frac{4 a^2 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{15 b^4 c^6}+\frac{4 \left (a+b \left (c x^3\right )^{3/2}\right )^{9/2}}{81 b^4 c^6}-\frac{4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{7/2}}{21 b^4 c^6} \]

[Out]

(-4*a^3*(a + b*(c*x^3)^(3/2))^(3/2))/(27*b^4*c^6) + (4*a^2*(a + b*(c*x^3)^(3/2))
^(5/2))/(15*b^4*c^6) - (4*a*(a + b*(c*x^3)^(3/2))^(7/2))/(21*b^4*c^6) + (4*(a +
b*(c*x^3)^(3/2))^(9/2))/(81*b^4*c^6)

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Rubi [A]  time = 0.16921, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{4 a^3 \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^4 c^6}+\frac{4 a^2 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{15 b^4 c^6}+\frac{4 \left (a+b \left (c x^3\right )^{3/2}\right )^{9/2}}{81 b^4 c^6}-\frac{4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{7/2}}{21 b^4 c^6} \]

Antiderivative was successfully verified.

[In]  Int[x^17*Sqrt[a + b*(c*x^3)^(3/2)],x]

[Out]

(-4*a^3*(a + b*(c*x^3)^(3/2))^(3/2))/(27*b^4*c^6) + (4*a^2*(a + b*(c*x^3)^(3/2))
^(5/2))/(15*b^4*c^6) - (4*a*(a + b*(c*x^3)^(3/2))^(7/2))/(21*b^4*c^6) + (4*(a +
b*(c*x^3)^(3/2))^(9/2))/(81*b^4*c^6)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{17} \sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**17*(a+b*(c*x**3)**(3/2))**(1/2),x)

[Out]

Integral(x**17*sqrt(a + b*(c*x**3)**(3/2)), x)

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Mathematica [A]  time = 0.105557, size = 0, normalized size = 0. \[ \int x^{17} \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[x^17*Sqrt[a + b*(c*x^3)^(3/2)],x]

[Out]

Integrate[x^17*Sqrt[a + b*(c*x^3)^(3/2)], x]

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{x}^{17}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^17*(a+b*(c*x^3)^(3/2))^(1/2),x)

[Out]

int(x^17*(a+b*(c*x^3)^(3/2))^(1/2),x)

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Maxima [A]  time = 1.34232, size = 115, normalized size = 0.99 \[ \frac{4 \,{\left (\frac{35 \,{\left (\left (c x^{3}\right )^{\frac{3}{2}} b + a\right )}^{\frac{9}{2}}}{b^{4}} - \frac{135 \,{\left (\left (c x^{3}\right )^{\frac{3}{2}} b + a\right )}^{\frac{7}{2}} a}{b^{4}} + \frac{189 \,{\left (\left (c x^{3}\right )^{\frac{3}{2}} b + a\right )}^{\frac{5}{2}} a^{2}}{b^{4}} - \frac{105 \,{\left (\left (c x^{3}\right )^{\frac{3}{2}} b + a\right )}^{\frac{3}{2}} a^{3}}{b^{4}}\right )}}{2835 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)*x^17,x, algorithm="maxima")

[Out]

4/2835*(35*((c*x^3)^(3/2)*b + a)^(9/2)/b^4 - 135*((c*x^3)^(3/2)*b + a)^(7/2)*a/b
^4 + 189*((c*x^3)^(3/2)*b + a)^(5/2)*a^2/b^4 - 105*((c*x^3)^(3/2)*b + a)^(3/2)*a
^3/b^4)/c^6

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Fricas [A]  time = 2.30362, size = 117, normalized size = 1.01 \[ \frac{4 \,{\left (35 \, b^{4} c^{6} x^{18} - 6 \, a^{2} b^{2} c^{3} x^{9} - 16 \, a^{4} +{\left (5 \, a b^{3} c^{4} x^{12} + 8 \, a^{3} b c x^{3}\right )} \sqrt{c x^{3}}\right )} \sqrt{\sqrt{c x^{3}} b c x^{3} + a}}{2835 \, b^{4} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)*x^17,x, algorithm="fricas")

[Out]

4/2835*(35*b^4*c^6*x^18 - 6*a^2*b^2*c^3*x^9 - 16*a^4 + (5*a*b^3*c^4*x^12 + 8*a^3
*b*c*x^3)*sqrt(c*x^3))*sqrt(sqrt(c*x^3)*b*c*x^3 + a)/(b^4*c^6)

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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**17*(a+b*(c*x**3)**(3/2))**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222591, size = 193, normalized size = 1.66 \[ \frac{4 \,{\left (\frac{16 \, \sqrt{a c^{3}} a^{4}}{b^{4} c^{5}} - \frac{105 \,{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{3}{2}} a^{3} c^{9} - 189 \,{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{5}{2}} a^{2} c^{6} + 135 \,{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{7}{2}} a c^{3} - 35 \,{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{9}{2}}}{b^{4} c^{17}}\right )}{\left | c \right |}}{2835 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)*x^17,x, algorithm="giac")

[Out]

4/2835*(16*sqrt(a*c^3)*a^4/(b^4*c^5) - (105*(sqrt(c*x)*b*c^4*x^4 + a*c^3)^(3/2)*
a^3*c^9 - 189*(sqrt(c*x)*b*c^4*x^4 + a*c^3)^(5/2)*a^2*c^6 + 135*(sqrt(c*x)*b*c^4
*x^4 + a*c^3)^(7/2)*a*c^3 - 35*(sqrt(c*x)*b*c^4*x^4 + a*c^3)^(9/2))/(b^4*c^17))*
abs(c)/c^(7/2)