3.1.55 \(\int x^7 (a+b x^2)^5 \, dx\) [55]

Optimal. Leaf size=72 \[ -\frac {a^3 \left (a+b x^2\right )^6}{12 b^4}+\frac {3 a^2 \left (a+b x^2\right )^7}{14 b^4}-\frac {3 a \left (a+b x^2\right )^8}{16 b^4}+\frac {\left (a+b x^2\right )^9}{18 b^4} \]

[Out]

-1/12*a^3*(b*x^2+a)^6/b^4+3/14*a^2*(b*x^2+a)^7/b^4-3/16*a*(b*x^2+a)^8/b^4+1/18*(b*x^2+a)^9/b^4

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Rubi [A]
time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} -\frac {a^3 \left (a+b x^2\right )^6}{12 b^4}+\frac {3 a^2 \left (a+b x^2\right )^7}{14 b^4}+\frac {\left (a+b x^2\right )^9}{18 b^4}-\frac {3 a \left (a+b x^2\right )^8}{16 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^2)^5,x]

[Out]

-1/12*(a^3*(a + b*x^2)^6)/b^4 + (3*a^2*(a + b*x^2)^7)/(14*b^4) - (3*a*(a + b*x^2)^8)/(16*b^4) + (a + b*x^2)^9/
(18*b^4)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^7 \left (a+b x^2\right )^5 \, dx &=\frac {1}{2} \text {Subst}\left (\int x^3 (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^5}{b^3}+\frac {3 a^2 (a+b x)^6}{b^3}-\frac {3 a (a+b x)^7}{b^3}+\frac {(a+b x)^8}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^3 \left (a+b x^2\right )^6}{12 b^4}+\frac {3 a^2 \left (a+b x^2\right )^7}{14 b^4}-\frac {3 a \left (a+b x^2\right )^8}{16 b^4}+\frac {\left (a+b x^2\right )^9}{18 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 69, normalized size = 0.96 \begin {gather*} \frac {a^5 x^8}{8}+\frac {1}{2} a^4 b x^{10}+\frac {5}{6} a^3 b^2 x^{12}+\frac {5}{7} a^2 b^3 x^{14}+\frac {5}{16} a b^4 x^{16}+\frac {b^5 x^{18}}{18} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^2)^5,x]

[Out]

(a^5*x^8)/8 + (a^4*b*x^10)/2 + (5*a^3*b^2*x^12)/6 + (5*a^2*b^3*x^14)/7 + (5*a*b^4*x^16)/16 + (b^5*x^18)/18

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Maple [A]
time = 0.04, size = 58, normalized size = 0.81

method	result	size
gosper	\(\frac {1}{8} a^{5} x^{8}+\frac {1}{2} a^{4} b \,x^{10}+\frac {5}{6} a^{3} b^{2} x^{12}+\frac {5}{7} a^{2} b^{3} x^{14}+\frac {5}{16} a \,b^{4} x^{16}+\frac {1}{18} b^{5} x^{18}\)	\(58\)
default	\(\frac {1}{8} a^{5} x^{8}+\frac {1}{2} a^{4} b \,x^{10}+\frac {5}{6} a^{3} b^{2} x^{12}+\frac {5}{7} a^{2} b^{3} x^{14}+\frac {5}{16} a \,b^{4} x^{16}+\frac {1}{18} b^{5} x^{18}\)	\(58\)
norman	\(\frac {1}{8} a^{5} x^{8}+\frac {1}{2} a^{4} b \,x^{10}+\frac {5}{6} a^{3} b^{2} x^{12}+\frac {5}{7} a^{2} b^{3} x^{14}+\frac {5}{16} a \,b^{4} x^{16}+\frac {1}{18} b^{5} x^{18}\)	\(58\)
risch	\(\frac {1}{8} a^{5} x^{8}+\frac {1}{2} a^{4} b \,x^{10}+\frac {5}{6} a^{3} b^{2} x^{12}+\frac {5}{7} a^{2} b^{3} x^{14}+\frac {5}{16} a \,b^{4} x^{16}+\frac {1}{18} b^{5} x^{18}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x^2+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/8*a^5*x^8+1/2*a^4*b*x^10+5/6*a^3*b^2*x^12+5/7*a^2*b^3*x^14+5/16*a*b^4*x^16+1/18*b^5*x^18

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Maxima [A]
time = 0.27, size = 57, normalized size = 0.79 \begin {gather*} \frac {1}{18} \, b^{5} x^{18} + \frac {5}{16} \, a b^{4} x^{16} + \frac {5}{7} \, a^{2} b^{3} x^{14} + \frac {5}{6} \, a^{3} b^{2} x^{12} + \frac {1}{2} \, a^{4} b x^{10} + \frac {1}{8} \, a^{5} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^5,x, algorithm="maxima")

[Out]

1/18*b^5*x^18 + 5/16*a*b^4*x^16 + 5/7*a^2*b^3*x^14 + 5/6*a^3*b^2*x^12 + 1/2*a^4*b*x^10 + 1/8*a^5*x^8

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Fricas [A]
time = 0.99, size = 57, normalized size = 0.79 \begin {gather*} \frac {1}{18} \, b^{5} x^{18} + \frac {5}{16} \, a b^{4} x^{16} + \frac {5}{7} \, a^{2} b^{3} x^{14} + \frac {5}{6} \, a^{3} b^{2} x^{12} + \frac {1}{2} \, a^{4} b x^{10} + \frac {1}{8} \, a^{5} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^5,x, algorithm="fricas")

[Out]

1/18*b^5*x^18 + 5/16*a*b^4*x^16 + 5/7*a^2*b^3*x^14 + 5/6*a^3*b^2*x^12 + 1/2*a^4*b*x^10 + 1/8*a^5*x^8

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Sympy [A]
time = 0.01, size = 65, normalized size = 0.90 \begin {gather*} \frac {a^{5} x^{8}}{8} + \frac {a^{4} b x^{10}}{2} + \frac {5 a^{3} b^{2} x^{12}}{6} + \frac {5 a^{2} b^{3} x^{14}}{7} + \frac {5 a b^{4} x^{16}}{16} + \frac {b^{5} x^{18}}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b*x**2+a)**5,x)

[Out]

a**5*x**8/8 + a**4*b*x**10/2 + 5*a**3*b**2*x**12/6 + 5*a**2*b**3*x**14/7 + 5*a*b**4*x**16/16 + b**5*x**18/18

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Giac [A]
time = 0.49, size = 57, normalized size = 0.79 \begin {gather*} \frac {1}{18} \, b^{5} x^{18} + \frac {5}{16} \, a b^{4} x^{16} + \frac {5}{7} \, a^{2} b^{3} x^{14} + \frac {5}{6} \, a^{3} b^{2} x^{12} + \frac {1}{2} \, a^{4} b x^{10} + \frac {1}{8} \, a^{5} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^5,x, algorithm="giac")

[Out]

1/18*b^5*x^18 + 5/16*a*b^4*x^16 + 5/7*a^2*b^3*x^14 + 5/6*a^3*b^2*x^12 + 1/2*a^4*b*x^10 + 1/8*a^5*x^8

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Mupad [B]
time = 0.02, size = 57, normalized size = 0.79 \begin {gather*} \frac {a^5\,x^8}{8}+\frac {a^4\,b\,x^{10}}{2}+\frac {5\,a^3\,b^2\,x^{12}}{6}+\frac {5\,a^2\,b^3\,x^{14}}{7}+\frac {5\,a\,b^4\,x^{16}}{16}+\frac {b^5\,x^{18}}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*x^2)^5,x)

[Out]

(a^5*x^8)/8 + (b^5*x^18)/18 + (a^4*b*x^10)/2 + (5*a*b^4*x^16)/16 + (5*a^3*b^2*x^12)/6 + (5*a^2*b^3*x^14)/7

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