3.4.87 \(\int x^7 (a+b x^2)^{5/2} \, dx\) [387]

Optimal. Leaf size=80 \[ -\frac {a^3 \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac {a^2 \left (a+b x^2\right )^{9/2}}{3 b^4}-\frac {3 a \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac {\left (a+b x^2\right )^{13/2}}{13 b^4} \]

[Out]

-1/7*a^3*(b*x^2+a)^(7/2)/b^4+1/3*a^2*(b*x^2+a)^(9/2)/b^4-3/11*a*(b*x^2+a)^(11/2)/b^4+1/13*(b*x^2+a)^(13/2)/b^4

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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \begin {gather*} -\frac {a^3 \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac {a^2 \left (a+b x^2\right )^{9/2}}{3 b^4}+\frac {\left (a+b x^2\right )^{13/2}}{13 b^4}-\frac {3 a \left (a+b x^2\right )^{11/2}}{11 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^2)^(5/2),x]

[Out]

-1/7*(a^3*(a + b*x^2)^(7/2))/b^4 + (a^2*(a + b*x^2)^(9/2))/(3*b^4) - (3*a*(a + b*x^2)^(11/2))/(11*b^4) + (a +
b*x^2)^(13/2)/(13*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 )^{5/2} \, dx &=\frac {1}{2} \text {Subst}\left (\int x^3 (a+b x)^{5/2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{5/2}}{b^3}+\frac {3 a^2 (a+b x)^{7/2}}{b^3}-\frac {3 a (a+b x)^{9/2}}{b^3}+\frac {(a+b x)^{11/2}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^3 \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac {a^2 \left (a+b x^2\right )^{9/2}}{3 b^4}-\frac {3 a \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac {\left (a+b x^2\right )^{13/2}}{13 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.62 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (-16 a^3+56 a^2 b x^2-126 a b^2 x^4+231 b^3 x^6\right )}{3003 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^2)^(5/2),x]

[Out]

((a + b*x^2)^(7/2)*(-16*a^3 + 56*a^2*b*x^2 - 126*a*b^2*x^4 + 231*b^3*x^6))/(3003*b^4)

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Maple [A]
time = 0.05, size = 82, normalized size = 1.02

method	result	size
gosper	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-231 b^{3} x^{6}+126 a \,b^{2} x^{4}-56 a^{2} b \,x^{2}+16 a^{3}\right )}{3003 b^{4}}\)	\(47\)
trager	\(-\frac {\left (-231 b^{6} x^{12}-567 a \,b^{5} x^{10}-371 a^{2} b^{4} x^{8}-5 a^{3} x^{6} b^{3}+6 a^{4} b^{2} x^{4}-8 a^{5} b \,x^{2}+16 a^{6}\right ) \sqrt {b \,x^{2}+a}}{3003 b^{4}}\)	\(80\)
risch	\(-\frac {\left (-231 b^{6} x^{12}-567 a \,b^{5} x^{10}-371 a^{2} b^{4} x^{8}-5 a^{3} x^{6} b^{3}+6 a^{4} b^{2} x^{4}-8 a^{5} b \,x^{2}+16 a^{6}\right ) \sqrt {b \,x^{2}+a}}{3003 b^{4}}\)	\(80\)
default	\(\frac {x^{6} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 b}-\frac {6 a \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )}{11 b}\right )}{13 b}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^6*(b*x^2+a)^(7/2)/b-6/13*a/b*(1/11*x^4*(b*x^2+a)^(7/2)/b-4/11*a/b*(1/9*x^2*(b*x^2+a)^(7/2)/b-2/63*a*(b*
x^2+a)^(7/2)/b^2))

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Maxima [A]
time = 0.29, size = 73, normalized size = 0.91 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} x^{6}}{13 \, b} - \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{4}}{143 \, b^{2}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{2}}{429 \, b^{3}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}}{3003 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*(b*x^2 + a)^(7/2)*x^6/b - 6/143*(b*x^2 + a)^(7/2)*a*x^4/b^2 + 8/429*(b*x^2 + a)^(7/2)*a^2*x^2/b^3 - 16/30
03*(b*x^2 + a)^(7/2)*a^3/b^4

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Fricas [A]
time = 1.31, size = 79, normalized size = 0.99 \begin {gather*} \frac {{\left (231 \, b^{6} x^{12} + 567 \, a b^{5} x^{10} + 371 \, a^{2} b^{4} x^{8} + 5 \, a^{3} b^{3} x^{6} - 6 \, a^{4} b^{2} x^{4} + 8 \, a^{5} b x^{2} - 16 \, a^{6}\right )} \sqrt {b x^{2} + a}}{3003 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3003*(231*b^6*x^12 + 567*a*b^5*x^10 + 371*a^2*b^4*x^8 + 5*a^3*b^3*x^6 - 6*a^4*b^2*x^4 + 8*a^5*b*x^2 - 16*a^6
)*sqrt(b*x^2 + a)/b^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (70) = 140\).
time = 0.63, size = 158, normalized size = 1.98 \begin {gather*} \begin {cases} - \frac {16 a^{6} \sqrt {a + b x^{2}}}{3003 b^{4}} + \frac {8 a^{5} x^{2} \sqrt {a + b x^{2}}}{3003 b^{3}} - \frac {2 a^{4} x^{4} \sqrt {a + b x^{2}}}{1001 b^{2}} + \frac {5 a^{3} x^{6} \sqrt {a + b x^{2}}}{3003 b} + \frac {53 a^{2} x^{8} \sqrt {a + b x^{2}}}{429} + \frac {27 a b x^{10} \sqrt {a + b x^{2}}}{143} + \frac {b^{2} x^{12} \sqrt {a + b x^{2}}}{13} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{2}} x^{8}}{8} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-16*a**6*sqrt(a + b*x**2)/(3003*b**4) + 8*a**5*x**2*sqrt(a + b*x**2)/(3003*b**3) - 2*a**4*x**4*sqrt
(a + b*x**2)/(1001*b**2) + 5*a**3*x**6*sqrt(a + b*x**2)/(3003*b) + 53*a**2*x**8*sqrt(a + b*x**2)/429 + 27*a*b*
x**10*sqrt(a + b*x**2)/143 + b**2*x**12*sqrt(a + b*x**2)/13, Ne(b, 0)), (a**(5/2)*x**8/8, True))

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Giac [A]
time = 1.17, size = 57, normalized size = 0.71 \begin {gather*} \frac {231 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} - 819 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a + 1001 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} a^{2} - 429 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}}{3003 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3003*(231*(b*x^2 + a)^(13/2) - 819*(b*x^2 + a)^(11/2)*a + 1001*(b*x^2 + a)^(9/2)*a^2 - 429*(b*x^2 + a)^(7/2)
*a^3)/b^4

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Mupad [B]
time = 4.86, size = 75, normalized size = 0.94 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {53\,a^2\,x^8}{429}-\frac {16\,a^6}{3003\,b^4}+\frac {b^2\,x^{12}}{13}+\frac {5\,a^3\,x^6}{3003\,b}-\frac {2\,a^4\,x^4}{1001\,b^2}+\frac {8\,a^5\,x^2}{3003\,b^3}+\frac {27\,a\,b\,x^{10}}{143}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*x^2)^(5/2),x)

[Out]

(a + b*x^2)^(1/2)*((53*a^2*x^8)/429 - (16*a^6)/(3003*b^4) + (b^2*x^12)/13 + (5*a^3*x^6)/(3003*b) - (2*a^4*x^4)
/(1001*b^2) + (8*a^5*x^2)/(3003*b^3) + (27*a*b*x^10)/143)

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