\(\int x^3 (a^2+2 a b x^3+b^2 x^6)^{3/2} \, dx\) [29]

Optimal result

Integrand size = 26, antiderivative size = 167 \[ \int x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {a^3 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {3 a^2 b x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {3 a b^2 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac {b^3 x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 276} \[ \int x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {3 a b^2 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac {3 a^2 b x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {b^3 x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {a^3 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]

[In]

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

[Out]

(a^3*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (3*a^2*b*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(
a + b*x^3)) + (3*a*b^2*x^10*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(10*(a + b*x^3)) + (b^3*x^13*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(13*(a + b*x^3))

Rule 276

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]

Rule 1369

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^3 \left (a b+b^2 x^3\right )^3 \, dx}{b^2 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a^3 b^3 x^3+3 a^2 b^4 x^6+3 a b^5 x^9+b^6 x^{12}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )} \\ & = \frac {a^3 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {3 a^2 b x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {3 a b^2 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac {b^3 x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37 \[ \int x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {x^4 \sqrt {\left (a+b x^3\right )^2} \left (455 a^3+780 a^2 b x^3+546 a b^2 x^6+140 b^3 x^9\right )}{1820 \left (a+b x^3\right )} \]

[In]

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

[Out]

(x^4*Sqrt[(a + b*x^3)^2]*(455*a^3 + 780*a^2*b*x^3 + 546*a*b^2*x^6 + 140*b^3*x^9))/(1820*(a + b*x^3))

Maple [A] (verified)

Time = 3.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.35

method	result	size
gosper	\(\frac {x^{4} \left (140 b^{3} x^{9}+546 b^{2} x^{6} a +780 a^{2} b \,x^{3}+455 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{1820 \left (b \,x^{3}+a \right )^{3}}\)	\(58\)
default	\(\frac {x^{4} \left (140 b^{3} x^{9}+546 b^{2} x^{6} a +780 a^{2} b \,x^{3}+455 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{1820 \left (b \,x^{3}+a \right )^{3}}\)	\(58\)
risch	\(\frac {a^{3} x^{4} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{4 b \,x^{3}+4 a}+\frac {3 a^{2} b \,x^{7} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{7 \left (b \,x^{3}+a \right )}+\frac {3 a \,b^{2} x^{10} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{10 \left (b \,x^{3}+a \right )}+\frac {b^{3} x^{13} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{13 b \,x^{3}+13 a}\)	\(116\)

[In]

int(x^3*(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/1820*x^4*(140*b^3*x^9+546*a*b^2*x^6+780*a^2*b*x^3+455*a^3)*((b*x^3+a)^2)^(3/2)/(b*x^3+a)^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.21 \[ \int x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{13} \, b^{3} x^{13} + \frac {3}{10} \, a b^{2} x^{10} + \frac {3}{7} \, a^{2} b x^{7} + \frac {1}{4} \, a^{3} x^{4} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\int x^{3} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.21 \[ \int x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{13} \, b^{3} x^{13} + \frac {3}{10} \, a b^{2} x^{10} + \frac {3}{7} \, a^{2} b x^{7} + \frac {1}{4} \, a^{3} x^{4} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.40 \[ \int x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{13} \, b^{3} x^{13} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{10} \, a b^{2} x^{10} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{7} \, a^{2} b x^{7} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{4} \, a^{3} x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\int x^3\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2} \,d x \]

[In]

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

[Out]

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