3.249 \(\int x^7 (c \sqrt{a+b x^2})^{3/2} \, dx\)

Optimal. Leaf size=138 \[ \frac{6 a^2 \left (a+b x^2\right )^2 \left (c \sqrt{a+b x^2}\right )^{3/2}}{11 b^4}-\frac{2 a^3 \left (a+b x^2\right ) \left (c \sqrt{a+b x^2}\right )^{3/2}}{7 b^4}+\frac{2 \left (a+b x^2\right )^4 \left (c \sqrt{a+b x^2}\right )^{3/2}}{19 b^4}-\frac{2 a \left (a+b x^2\right )^3 \left (c \sqrt{a+b x^2}\right )^{3/2}}{5 b^4} \]

[Out]

(-2*a^3*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2))/(7*b^4) + (6*a^2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)^2)/(11*b
^4) - (2*a*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)^3)/(5*b^4) + (2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)^4)/(19*
b^4)

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Rubi [A]  time = 0.1879, antiderivative size = 152, normalized size of antiderivative = 1.1, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6720, 266, 43} \[ \frac{6 a^2 c \left (a+b x^2\right )^{5/2} \sqrt{c \sqrt{a+b x^2}}}{11 b^4}-\frac{2 a^3 c \left (a+b x^2\right )^{3/2} \sqrt{c \sqrt{a+b x^2}}}{7 b^4}+\frac{2 c \left (a+b x^2\right )^{9/2} \sqrt{c \sqrt{a+b x^2}}}{19 b^4}-\frac{2 a c \left (a+b x^2\right )^{7/2} \sqrt{c \sqrt{a+b x^2}}}{5 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(c*Sqrt[a + b*x^2])^(3/2),x]

[Out]

(-2*a^3*c*Sqrt[c*Sqrt[a + b*x^2]]*(a + b*x^2)^(3/2))/(7*b^4) + (6*a^2*c*Sqrt[c*Sqrt[a + b*x^2]]*(a + b*x^2)^(5
/2))/(11*b^4) - (2*a*c*Sqrt[c*Sqrt[a + b*x^2]]*(a + b*x^2)^(7/2))/(5*b^4) + (2*c*Sqrt[c*Sqrt[a + b*x^2]]*(a +
b*x^2)^(9/2))/(19*b^4)

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 266

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]]

Rule 43

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])

Rubi steps

\begin{align*} \int x^7 \left (c \sqrt{a+b x^2}\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \int x^7 \left (a+b x^2\right )^{3/4} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \operatorname{Subst}\left (\int x^3 (a+b x)^{3/4} \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{3/4}}{b^3}+\frac{3 a^2 (a+b x)^{7/4}}{b^3}-\frac{3 a (a+b x)^{11/4}}{b^3}+\frac{(a+b x)^{15/4}}{b^3}\right ) \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=-\frac{2 a^3 c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{3/2}}{7 b^4}+\frac{6 a^2 c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{5/2}}{11 b^4}-\frac{2 a c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{7/2}}{5 b^4}+\frac{2 c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{9/2}}{19 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0372595, size = 63, normalized size = 0.46 \[ \frac{2 \left (a+b x^2\right ) \left (224 a^2 b x^2-128 a^3-308 a b^2 x^4+385 b^3 x^6\right ) \left (c \sqrt{a+b x^2}\right )^{3/2}}{7315 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(c*Sqrt[a + b*x^2])^(3/2),x]

[Out]

(2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)*(-128*a^3 + 224*a^2*b*x^2 - 308*a*b^2*x^4 + 385*b^3*x^6))/(7315*b^4)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 2\,b{x}^{2}+2\,a \right ) \left ( -385\,{b}^{3}{x}^{6}+308\,a{b}^{2}{x}^{4}-224\,{a}^{2}b{x}^{2}+128\,{a}^{3} \right ) }{7315\,{b}^{4}} \left ( c\sqrt{b{x}^{2}+a} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(c*(b*x^2+a)^(1/2))^(3/2),x)

[Out]

-2/7315*(b*x^2+a)*(-385*b^3*x^6+308*a*b^2*x^4-224*a^2*b*x^2+128*a^3)*(c*(b*x^2+a)^(1/2))^(3/2)/b^4

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Maxima [A]  time = 1.0125, size = 115, normalized size = 0.83 \begin{align*} -\frac{2 \,{\left (1045 \, \left (\sqrt{b x^{2} + a} c\right )^{\frac{7}{2}} a^{3} c^{6} - 1995 \, \left (\sqrt{b x^{2} + a} c\right )^{\frac{11}{2}} a^{2} c^{4} + 1463 \, \left (\sqrt{b x^{2} + a} c\right )^{\frac{15}{2}} a c^{2} - 385 \, \left (\sqrt{b x^{2} + a} c\right )^{\frac{19}{2}}\right )}}{7315 \, b^{4} c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(c*(b*x^2+a)^(1/2))^(3/2),x, algorithm="maxima")

[Out]

-2/7315*(1045*(sqrt(b*x^2 + a)*c)^(7/2)*a^3*c^6 - 1995*(sqrt(b*x^2 + a)*c)^(11/2)*a^2*c^4 + 1463*(sqrt(b*x^2 +
a)*c)^(15/2)*a*c^2 - 385*(sqrt(b*x^2 + a)*c)^(19/2))/(b^4*c^8)

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Fricas [A]  time = 1.70495, size = 180, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (385 \, b^{4} c x^{8} + 77 \, a b^{3} c x^{6} - 84 \, a^{2} b^{2} c x^{4} + 96 \, a^{3} b c x^{2} - 128 \, a^{4} c\right )} \sqrt{b x^{2} + a} \sqrt{\sqrt{b x^{2} + a} c}}{7315 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(c*(b*x^2+a)^(1/2))^(3/2),x, algorithm="fricas")

[Out]

2/7315*(385*b^4*c*x^8 + 77*a*b^3*c*x^6 - 84*a^2*b^2*c*x^4 + 96*a^3*b*c*x^2 - 128*a^4*c)*sqrt(b*x^2 + a)*sqrt(s
qrt(b*x^2 + a)*c)/b^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(c*(b*x**2+a)**(1/2))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16246, size = 81, normalized size = 0.59 \begin{align*} \frac{2 \,{\left (385 \,{\left (b x^{2} + a\right )}^{\frac{19}{4}} - 1463 \,{\left (b x^{2} + a\right )}^{\frac{15}{4}} a + 1995 \,{\left (b x^{2} + a\right )}^{\frac{11}{4}} a^{2} - 1045 \,{\left (b x^{2} + a\right )}^{\frac{7}{4}} a^{3}\right )} c^{\frac{3}{2}}}{7315 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(c*(b*x^2+a)^(1/2))^(3/2),x, algorithm="giac")

[Out]

2/7315*(385*(b*x^2 + a)^(19/4) - 1463*(b*x^2 + a)^(15/4)*a + 1995*(b*x^2 + a)^(11/4)*a^2 - 1045*(b*x^2 + a)^(7
/4)*a^3)*c^(3/2)/b^4