3.5.46 \(\int x^6 (a^2+2 a b x^2+b^2 x^4)^3 \, dx\) [446]

Optimal. Leaf size=79 \[ \frac {a^6 x^7}{7}+\frac {2}{3} a^5 b x^9+\frac {15}{11} a^4 b^2 x^{11}+\frac {20}{13} a^3 b^3 x^{13}+a^2 b^4 x^{15}+\frac {6}{17} a b^5 x^{17}+\frac {b^6 x^{19}}{19} \]

[Out]

1/7*a^6*x^7+2/3*a^5*b*x^9+15/11*a^4*b^2*x^11+20/13*a^3*b^3*x^13+a^2*b^4*x^15+6/17*a*b^5*x^17+1/19*b^6*x^19

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Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 276} \begin {gather*} \frac {a^6 x^7}{7}+\frac {2}{3} a^5 b x^9+\frac {15}{11} a^4 b^2 x^{11}+\frac {20}{13} a^3 b^3 x^{13}+a^2 b^4 x^{15}+\frac {6}{17} a b^5 x^{17}+\frac {b^6 x^{19}}{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^7)/7 + (2*a^5*b*x^9)/3 + (15*a^4*b^2*x^11)/11 + (20*a^3*b^3*x^13)/13 + a^2*b^4*x^15 + (6*a*b^5*x^17)/17
+ (b^6*x^19)/19

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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]

Rubi steps

\begin {align*} \int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int x^6 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\int \left (a^6 b^6 x^6+6 a^5 b^7 x^8+15 a^4 b^8 x^{10}+20 a^3 b^9 x^{12}+15 a^2 b^{10} x^{14}+6 a b^{11} x^{16}+b^{12} x^{18}\right ) \, dx}{b^6}\\ &=\frac {a^6 x^7}{7}+\frac {2}{3} a^5 b x^9+\frac {15}{11} a^4 b^2 x^{11}+\frac {20}{13} a^3 b^3 x^{13}+a^2 b^4 x^{15}+\frac {6}{17} a b^5 x^{17}+\frac {b^6 x^{19}}{19}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 79, normalized size = 1.00 \begin {gather*} \frac {a^6 x^7}{7}+\frac {2}{3} a^5 b x^9+\frac {15}{11} a^4 b^2 x^{11}+\frac {20}{13} a^3 b^3 x^{13}+a^2 b^4 x^{15}+\frac {6}{17} a b^5 x^{17}+\frac {b^6 x^{19}}{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^7)/7 + (2*a^5*b*x^9)/3 + (15*a^4*b^2*x^11)/11 + (20*a^3*b^3*x^13)/13 + a^2*b^4*x^15 + (6*a*b^5*x^17)/17
+ (b^6*x^19)/19

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Maple [A]
time = 0.05, size = 68, normalized size = 0.86

method	result	size
default	\(\frac {1}{7} a^{6} x^{7}+\frac {2}{3} a^{5} b \,x^{9}+\frac {15}{11} a^{4} b^{2} x^{11}+\frac {20}{13} a^{3} b^{3} x^{13}+a^{2} b^{4} x^{15}+\frac {6}{17} a \,b^{5} x^{17}+\frac {1}{19} b^{6} x^{19}\)	\(68\)
norman	\(\frac {1}{7} a^{6} x^{7}+\frac {2}{3} a^{5} b \,x^{9}+\frac {15}{11} a^{4} b^{2} x^{11}+\frac {20}{13} a^{3} b^{3} x^{13}+a^{2} b^{4} x^{15}+\frac {6}{17} a \,b^{5} x^{17}+\frac {1}{19} b^{6} x^{19}\)	\(68\)
risch	\(\frac {1}{7} a^{6} x^{7}+\frac {2}{3} a^{5} b \,x^{9}+\frac {15}{11} a^{4} b^{2} x^{11}+\frac {20}{13} a^{3} b^{3} x^{13}+a^{2} b^{4} x^{15}+\frac {6}{17} a \,b^{5} x^{17}+\frac {1}{19} b^{6} x^{19}\)	\(68\)
gosper	\(\frac {x^{7} \left (51051 b^{6} x^{12}+342342 a \,b^{5} x^{10}+969969 a^{2} b^{4} x^{8}+1492260 a^{3} b^{3} x^{6}+1322685 a^{4} b^{2} x^{4}+646646 a^{5} b \,x^{2}+138567 a^{6}\right )}{969969}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/7*a^6*x^7+2/3*a^5*b*x^9+15/11*a^4*b^2*x^11+20/13*a^3*b^3*x^13+a^2*b^4*x^15+6/17*a*b^5*x^17+1/19*b^6*x^19

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Maxima [A]
time = 0.35, size = 67, normalized size = 0.85 \begin {gather*} \frac {1}{19} \, b^{6} x^{19} + \frac {6}{17} \, a b^{5} x^{17} + a^{2} b^{4} x^{15} + \frac {20}{13} \, a^{3} b^{3} x^{13} + \frac {15}{11} \, a^{4} b^{2} x^{11} + \frac {2}{3} \, a^{5} b x^{9} + \frac {1}{7} \, a^{6} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/19*b^6*x^19 + 6/17*a*b^5*x^17 + a^2*b^4*x^15 + 20/13*a^3*b^3*x^13 + 15/11*a^4*b^2*x^11 + 2/3*a^5*b*x^9 + 1/7
*a^6*x^7

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Fricas [A]
time = 0.34, size = 67, normalized size = 0.85 \begin {gather*} \frac {1}{19} \, b^{6} x^{19} + \frac {6}{17} \, a b^{5} x^{17} + a^{2} b^{4} x^{15} + \frac {20}{13} \, a^{3} b^{3} x^{13} + \frac {15}{11} \, a^{4} b^{2} x^{11} + \frac {2}{3} \, a^{5} b x^{9} + \frac {1}{7} \, a^{6} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/19*b^6*x^19 + 6/17*a*b^5*x^17 + a^2*b^4*x^15 + 20/13*a^3*b^3*x^13 + 15/11*a^4*b^2*x^11 + 2/3*a^5*b*x^9 + 1/7
*a^6*x^7

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Sympy [A]
time = 0.01, size = 76, normalized size = 0.96 \begin {gather*} \frac {a^{6} x^{7}}{7} + \frac {2 a^{5} b x^{9}}{3} + \frac {15 a^{4} b^{2} x^{11}}{11} + \frac {20 a^{3} b^{3} x^{13}}{13} + a^{2} b^{4} x^{15} + \frac {6 a b^{5} x^{17}}{17} + \frac {b^{6} x^{19}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**6*x**7/7 + 2*a**5*b*x**9/3 + 15*a**4*b**2*x**11/11 + 20*a**3*b**3*x**13/13 + a**2*b**4*x**15 + 6*a*b**5*x**
17/17 + b**6*x**19/19

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Giac [A]
time = 4.27, size = 67, normalized size = 0.85 \begin {gather*} \frac {1}{19} \, b^{6} x^{19} + \frac {6}{17} \, a b^{5} x^{17} + a^{2} b^{4} x^{15} + \frac {20}{13} \, a^{3} b^{3} x^{13} + \frac {15}{11} \, a^{4} b^{2} x^{11} + \frac {2}{3} \, a^{5} b x^{9} + \frac {1}{7} \, a^{6} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/19*b^6*x^19 + 6/17*a*b^5*x^17 + a^2*b^4*x^15 + 20/13*a^3*b^3*x^13 + 15/11*a^4*b^2*x^11 + 2/3*a^5*b*x^9 + 1/7
*a^6*x^7

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Mupad [B]
time = 0.03, size = 67, normalized size = 0.85 \begin {gather*} \frac {a^6\,x^7}{7}+\frac {2\,a^5\,b\,x^9}{3}+\frac {15\,a^4\,b^2\,x^{11}}{11}+\frac {20\,a^3\,b^3\,x^{13}}{13}+a^2\,b^4\,x^{15}+\frac {6\,a\,b^5\,x^{17}}{17}+\frac {b^6\,x^{19}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

(a^6*x^7)/7 + (b^6*x^19)/19 + (2*a^5*b*x^9)/3 + (6*a*b^5*x^17)/17 + (15*a^4*b^2*x^11)/11 + (20*a^3*b^3*x^13)/1
3 + a^2*b^4*x^15

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