3.1.60 \(\int x^3 (a^2+2 a b x^3+b^2 x^6)^{5/2} \, dx\) [60]

Optimal. Leaf size=252 \[ \frac {a^5 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {5 a^4 b x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {a^3 b^2 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {10 a^2 b^3 x^{13} \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac {5 a b^4 x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac {b^5 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 252, normalized size of antiderivative = 1.00, 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 = {1369, 276} \begin {gather*} \frac {b^5 x^{19} \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 \left (a+b x^3\right )}+\frac {5 a b^4 x^{16} \sqrt {a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac {10 a^2 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^5 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {5 a^4 b x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {a^3 b^2 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.01, size = 83, normalized size = 0.33 \begin {gather*} \frac {x^4 \sqrt {\left (a+b x^3\right )^2} \left (6916 a^5+19760 a^4 b x^3+27664 a^3 b^2 x^6+21280 a^2 b^3 x^9+8645 a b^4 x^{12}+1456 b^5 x^{15}\right )}{27664 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Sqrt[(a + b*x^3)^2]*(6916*a^5 + 19760*a^4*b*x^3 + 27664*a^3*b^2*x^6 + 21280*a^2*b^3*x^9 + 8645*a*b^4*x^12
+ 1456*b^5*x^15))/(27664*(a + b*x^3))

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Maple [A]
time = 0.06, size = 80, normalized size = 0.32

method	result	size
gosper	\(\frac {x^{4} \left (1456 b^{5} x^{15}+8645 b^{4} a \,x^{12}+21280 a^{2} b^{3} x^{9}+27664 b^{2} a^{3} x^{6}+19760 a^{4} b \,x^{3}+6916 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{27664 \left (b \,x^{3}+a \right )^{5}}\)	\(80\)
default	\(\frac {x^{4} \left (1456 b^{5} x^{15}+8645 b^{4} a \,x^{12}+21280 a^{2} b^{3} x^{9}+27664 b^{2} a^{3} x^{6}+19760 a^{4} b \,x^{3}+6916 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{27664 \left (b \,x^{3}+a \right )^{5}}\)	\(80\)
risch	\(\frac {a^{5} x^{4} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{4 b \,x^{3}+4 a}+\frac {5 a^{4} b \,x^{7} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{7 \left (b \,x^{3}+a \right )}+\frac {a^{3} b^{2} x^{10} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{b \,x^{3}+a}+\frac {10 a^{2} b^{3} x^{13} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{13 \left (b \,x^{3}+a \right )}+\frac {5 a \,b^{4} x^{16} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{16 \left (b \,x^{3}+a \right )}+\frac {b^{5} x^{19} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{19 b \,x^{3}+19 a}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27664*x^4*(1456*b^5*x^15+8645*a*b^4*x^12+21280*a^2*b^3*x^9+27664*a^3*b^2*x^6+19760*a^4*b*x^3+6916*a^5)*((b*x
^3+a)^2)^(5/2)/(b*x^3+a)^5

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Maxima [A]
time = 0.29, size = 56, normalized size = 0.22 \begin {gather*} \frac {1}{19} \, b^{5} x^{19} + \frac {5}{16} \, a b^{4} x^{16} + \frac {10}{13} \, a^{2} b^{3} x^{13} + a^{3} b^{2} x^{10} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{4} \, a^{5} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 56, normalized size = 0.22 \begin {gather*} \frac {1}{19} \, b^{5} x^{19} + \frac {5}{16} \, a b^{4} x^{16} + \frac {10}{13} \, a^{2} b^{3} x^{13} + a^{3} b^{2} x^{10} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{4} \, a^{5} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 8.08, size = 104, normalized size = 0.41 \begin {gather*} \frac {1}{19} \, b^{5} x^{19} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{16} \, a b^{4} x^{16} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{13} \, a^{2} b^{3} x^{13} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{3} b^{2} x^{10} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{7} \, a^{4} b x^{7} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{4} \, a^{5} x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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