3.57 \(\int x^6 (a^2+2 a b x^3+b^2 x^6)^{5/2} \, dx\)

Optimal. Leaf size=255 \[ \frac{b^5 x^{22} \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 \left (a+b x^3\right )}+\frac{5 a b^4 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{a^4 b x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{a^5 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \]

[Out]

(a^5*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (a^4*b*x^10*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a
+ b*x^3)) + (10*a^3*b^2*x^13*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(13*(a + b*x^3)) + (5*a^2*b^3*x^16*Sqrt[a^2 + 2
*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (5*a*b^4*x^19*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(19*(a + b*x^3)) + (b^5*
x^22*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(22*(a + b*x^3))

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Rubi [A]  time = 0.0594273, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 270} \[ \frac{b^5 x^{22} \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 \left (a+b x^3\right )}+\frac{5 a b^4 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{a^4 b x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{a^5 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(a^5*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (a^4*b*x^10*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a
+ b*x^3)) + (10*a^3*b^2*x^13*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(13*(a + b*x^3)) + (5*a^2*b^3*x^16*Sqrt[a^2 + 2
*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (5*a*b^4*x^19*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(19*(a + b*x^3)) + (b^5*
x^22*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(22*(a + b*x^3))

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^6 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int x^6 \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (a^5 b^5 x^6+5 a^4 b^6 x^9+10 a^3 b^7 x^{12}+10 a^2 b^8 x^{15}+5 a b^9 x^{18}+b^{10} x^{21}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{a^5 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^4 b x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{5 a b^4 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac{b^5 x^{22} \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0207801, size = 83, normalized size = 0.33 \[ \frac{x^7 \sqrt{\left (a+b x^3\right )^2} \left (95095 a^2 b^3 x^9+117040 a^3 b^2 x^6+76076 a^4 b x^3+21736 a^5+40040 a b^4 x^{12}+6916 b^5 x^{15}\right )}{152152 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(x^7*Sqrt[(a + b*x^3)^2]*(21736*a^5 + 76076*a^4*b*x^3 + 117040*a^3*b^2*x^6 + 95095*a^2*b^3*x^9 + 40040*a*b^4*x
^12 + 6916*b^5*x^15))/(152152*(a + b*x^3))

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Maple [A]  time = 0.006, size = 80, normalized size = 0.3 \begin{align*}{\frac{{x}^{7} \left ( 6916\,{b}^{5}{x}^{15}+40040\,a{b}^{4}{x}^{12}+95095\,{a}^{2}{b}^{3}{x}^{9}+117040\,{a}^{3}{b}^{2}{x}^{6}+76076\,{a}^{4}b{x}^{3}+21736\,{a}^{5} \right ) }{152152\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/152152*x^7*(6916*b^5*x^15+40040*a*b^4*x^12+95095*a^2*b^3*x^9+117040*a^3*b^2*x^6+76076*a^4*b*x^3+21736*a^5)*(
(b*x^3+a)^2)^(5/2)/(b*x^3+a)^5

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Maxima [A]  time = 1.01154, size = 77, normalized size = 0.3 \begin{align*} \frac{1}{22} \, b^{5} x^{22} + \frac{5}{19} \, a b^{4} x^{19} + \frac{5}{8} \, a^{2} b^{3} x^{16} + \frac{10}{13} \, a^{3} b^{2} x^{13} + \frac{1}{2} \, a^{4} b x^{10} + \frac{1}{7} \, a^{5} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/22*b^5*x^22 + 5/19*a*b^4*x^19 + 5/8*a^2*b^3*x^16 + 10/13*a^3*b^2*x^13 + 1/2*a^4*b*x^10 + 1/7*a^5*x^7

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Fricas [A]  time = 1.75121, size = 140, normalized size = 0.55 \begin{align*} \frac{1}{22} \, b^{5} x^{22} + \frac{5}{19} \, a b^{4} x^{19} + \frac{5}{8} \, a^{2} b^{3} x^{16} + \frac{10}{13} \, a^{3} b^{2} x^{13} + \frac{1}{2} \, a^{4} b x^{10} + \frac{1}{7} \, a^{5} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/22*b^5*x^22 + 5/19*a*b^4*x^19 + 5/8*a^2*b^3*x^16 + 10/13*a^3*b^2*x^13 + 1/2*a^4*b*x^10 + 1/7*a^5*x^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**6*((a + b*x**3)**2)**(5/2), x)

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Giac [A]  time = 1.09437, size = 142, normalized size = 0.56 \begin{align*} \frac{1}{22} \, b^{5} x^{22} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{19} \, a b^{4} x^{19} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{8} \, a^{2} b^{3} x^{16} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{13} \, a^{3} b^{2} x^{13} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{2} \, a^{4} b x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{7} \, a^{5} x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

1/22*b^5*x^22*sgn(b*x^3 + a) + 5/19*a*b^4*x^19*sgn(b*x^3 + a) + 5/8*a^2*b^3*x^16*sgn(b*x^3 + a) + 10/13*a^3*b^
2*x^13*sgn(b*x^3 + a) + 1/2*a^4*b*x^10*sgn(b*x^3 + a) + 1/7*a^5*x^7*sgn(b*x^3 + a)