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

Optimal. Leaf size=247 \[ \frac{b^5 x^{16} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{16 \left (a+b x^3\right )^5}+\frac{5 a b^4 x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{13 \left (a+b x^3\right )^5}+\frac{a^2 b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac{10 a^3 b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{7 \left (a+b x^3\right )^5}+\frac{5 a^4 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{4 \left (a+b x^3\right )^5}+\frac{a^5 x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5} \]

[Out]

(a^5*x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(a + b*x^3)^5 + (5*a^4*b*x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(4*(
a + b*x^3)^5) + (10*a^3*b^2*x^7*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(7*(a + b*x^3)^5) + (a^2*b^3*x^10*(a^2 + 2*
a*b*x^3 + b^2*x^6)^(5/2))/(a + b*x^3)^5 + (5*a*b^4*x^13*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(13*(a + b*x^3)^5)
+ (b^5*x^16*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(16*(a + b*x^3)^5)

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Rubi [A]  time = 0.0508262, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1343, 194} \[ \frac{b^5 x^{16} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{16 \left (a+b x^3\right )^5}+\frac{5 a b^4 x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{13 \left (a+b x^3\right )^5}+\frac{a^2 b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac{10 a^3 b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{7 \left (a+b x^3\right )^5}+\frac{5 a^4 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{4 \left (a+b x^3\right )^5}+\frac{a^5 x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(a + b*x^3)^5 + (5*a^4*b*x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(4*(
a + b*x^3)^5) + (10*a^3*b^2*x^7*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(7*(a + b*x^3)^5) + (a^2*b^3*x^10*(a^2 + 2*
a*b*x^3 + b^2*x^6)^(5/2))/(a + b*x^3)^5 + (5*a*b^4*x^13*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(13*(a + b*x^3)^5)
+ (b^5*x^16*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(16*(a + b*x^3)^5)

Rule 1343

Int[((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x
^n)^(2*p), Int[(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0]

Rule 194

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

Rubi steps

\begin{align*} \int \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \int \left (2 a b+2 b^2 x^3\right )^5 \, dx}{\left (2 a b+2 b^2 x^3\right )^5}\\ &=\frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \int \left (32 a^5 b^5+160 a^4 b^6 x^3+320 a^3 b^7 x^6+320 a^2 b^8 x^9+160 a b^9 x^{12}+32 b^{10} x^{15}\right ) \, dx}{\left (2 a b+2 b^2 x^3\right )^5}\\ &=\frac{a^5 x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac{5 a^4 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{4 \left (a+b x^3\right )^5}+\frac{10 a^3 b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{7 \left (a+b x^3\right )^5}+\frac{a^2 b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac{5 a b^4 x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{13 \left (a+b x^3\right )^5}+\frac{b^5 x^{16} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{16 \left (a+b x^3\right )^5}\\ \end{align*}

Mathematica [A]  time = 0.0205875, size = 81, normalized size = 0.33 \[ \frac{x \sqrt{\left (a+b x^3\right )^2} \left (1456 a^2 b^3 x^9+2080 a^3 b^2 x^6+1820 a^4 b x^3+1456 a^5+560 a b^4 x^{12}+91 b^5 x^{15}\right )}{1456 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x^3)^2]*(1456*a^5 + 1820*a^4*b*x^3 + 2080*a^3*b^2*x^6 + 1456*a^2*b^3*x^9 + 560*a*b^4*x^12 + 91*
b^5*x^15))/(1456*(a + b*x^3))

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Maple [A]  time = 0.004, size = 78, normalized size = 0.3 \begin{align*}{\frac{x \left ( 91\,{b}^{5}{x}^{15}+560\,a{b}^{4}{x}^{12}+1456\,{a}^{2}{b}^{3}{x}^{9}+2080\,{a}^{3}{b}^{2}{x}^{6}+1820\,{a}^{4}b{x}^{3}+1456\,{a}^{5} \right ) }{1456\, \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((b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/1456*x*(91*b^5*x^15+560*a*b^4*x^12+1456*a^2*b^3*x^9+2080*a^3*b^2*x^6+1820*a^4*b*x^3+1456*a^5)*((b*x^3+a)^2)^
(5/2)/(b*x^3+a)^5

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Maxima [A]  time = 1.02702, size = 72, normalized size = 0.29 \begin{align*} \frac{1}{16} \, b^{5} x^{16} + \frac{5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{4} \, a^{4} b x^{4} + a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.76481, size = 123, normalized size = 0.5 \begin{align*} \frac{1}{16} \, b^{5} x^{16} + \frac{5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{4} \, a^{4} b x^{4} + a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11208, size = 136, normalized size = 0.55 \begin{align*} \frac{1}{16} \, b^{5} x^{16} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{13} \, a b^{4} x^{13} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{2} b^{3} x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{7} \, a^{3} b^{2} x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{4} \, a^{4} b x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{5} x \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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