3.425 \(\int x^6 (a+b x^2)^{9/2} \, dx\)

Optimal. Leaf size=202 \[ \frac{45 a^7 x \sqrt{a+b x^2}}{32768 b^3}-\frac{15 a^6 x^3 \sqrt{a+b x^2}}{16384 b^2}-\frac{45 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{32768 b^{7/2}}+\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2} \]

[Out]

(45*a^7*x*Sqrt[a + b*x^2])/(32768*b^3) - (15*a^6*x^3*Sqrt[a + b*x^2])/(16384*b^2) + (3*a^5*x^5*Sqrt[a + b*x^2]
)/(4096*b) + (9*a^4*x^7*Sqrt[a + b*x^2])/2048 + (3*a^3*x^7*(a + b*x^2)^(3/2))/256 + (3*a^2*x^7*(a + b*x^2)^(5/
2))/128 + (9*a*x^7*(a + b*x^2)^(7/2))/224 + (x^7*(a + b*x^2)^(9/2))/16 - (45*a^8*ArcTanh[(Sqrt[b]*x)/Sqrt[a +
b*x^2]])/(32768*b^(7/2))

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Rubi [A]  time = 0.11412, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \[ \frac{45 a^7 x \sqrt{a+b x^2}}{32768 b^3}-\frac{15 a^6 x^3 \sqrt{a+b x^2}}{16384 b^2}-\frac{45 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{32768 b^{7/2}}+\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x^2)^(9/2),x]

[Out]

(45*a^7*x*Sqrt[a + b*x^2])/(32768*b^3) - (15*a^6*x^3*Sqrt[a + b*x^2])/(16384*b^2) + (3*a^5*x^5*Sqrt[a + b*x^2]
)/(4096*b) + (9*a^4*x^7*Sqrt[a + b*x^2])/2048 + (3*a^3*x^7*(a + b*x^2)^(3/2))/256 + (3*a^2*x^7*(a + b*x^2)^(5/
2))/128 + (9*a*x^7*(a + b*x^2)^(7/2))/224 + (x^7*(a + b*x^2)^(9/2))/16 - (45*a^8*ArcTanh[(Sqrt[b]*x)/Sqrt[a +
b*x^2]])/(32768*b^(7/2))

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^6 \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac{1}{16} (9 a) \int x^6 \left (a+b x^2\right )^{7/2} \, dx\\ &=\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac{1}{32} \left (9 a^2\right ) \int x^6 \left (a+b x^2\right )^{5/2} \, dx\\ &=\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac{1}{128} \left (15 a^3\right ) \int x^6 \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac{1}{256} \left (9 a^4\right ) \int x^6 \sqrt{a+b x^2} \, dx\\ &=\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac{\left (9 a^5\right ) \int \frac{x^6}{\sqrt{a+b x^2}} \, dx}{2048}\\ &=\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}-\frac{\left (15 a^6\right ) \int \frac{x^4}{\sqrt{a+b x^2}} \, dx}{4096 b}\\ &=-\frac{15 a^6 x^3 \sqrt{a+b x^2}}{16384 b^2}+\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}+\frac{\left (45 a^7\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{16384 b^2}\\ &=\frac{45 a^7 x \sqrt{a+b x^2}}{32768 b^3}-\frac{15 a^6 x^3 \sqrt{a+b x^2}}{16384 b^2}+\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}-\frac{\left (45 a^8\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{32768 b^3}\\ &=\frac{45 a^7 x \sqrt{a+b x^2}}{32768 b^3}-\frac{15 a^6 x^3 \sqrt{a+b x^2}}{16384 b^2}+\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}-\frac{\left (45 a^8\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{32768 b^3}\\ &=\frac{45 a^7 x \sqrt{a+b x^2}}{32768 b^3}-\frac{15 a^6 x^3 \sqrt{a+b x^2}}{16384 b^2}+\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left (a+b x^2\right )^{3/2}+\frac{3}{128} a^2 x^7 \left (a+b x^2\right )^{5/2}+\frac{9}{224} a x^7 \left (a+b x^2\right )^{7/2}+\frac{1}{16} x^7 \left (a+b x^2\right )^{9/2}-\frac{45 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{32768 b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.213618, size = 138, normalized size = 0.68 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (119040 a^2 b^5 x^{10}+98432 a^3 b^4 x^8+32624 a^4 b^3 x^6+168 a^5 b^2 x^4-210 a^6 b x^2+315 a^7+66560 a b^6 x^{12}+14336 b^7 x^{14}\right )-\frac{315 a^{15/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{229376 b^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x^2)^(9/2),x]

[Out]

(Sqrt[a + b*x^2]*(Sqrt[b]*x*(315*a^7 - 210*a^6*b*x^2 + 168*a^5*b^2*x^4 + 32624*a^4*b^3*x^6 + 98432*a^3*b^4*x^8
+ 119040*a^2*b^5*x^10 + 66560*a*b^6*x^12 + 14336*b^7*x^14) - (315*a^(15/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt
[1 + (b*x^2)/a]))/(229376*b^(7/2))

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Maple [A]  time = 0.011, size = 169, normalized size = 0.8 \begin{align*}{\frac{{x}^{5}}{16\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{5\,a{x}^{3}}{224\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{5\,{a}^{2}x}{896\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{a}^{3}x}{1792\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}-{\frac{9\,{a}^{4}x}{14336\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{a}^{5}x}{4096\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{15\,{a}^{6}x}{16384\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{45\,{a}^{7}x}{32768\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{45\,{a}^{8}}{32768}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x^2+a)^(9/2),x)

[Out]

1/16*x^5*(b*x^2+a)^(11/2)/b-5/224/b^2*a*x^3*(b*x^2+a)^(11/2)+5/896/b^3*a^2*x*(b*x^2+a)^(11/2)-1/1792/b^3*a^3*x
*(b*x^2+a)^(9/2)-9/14336/b^3*a^4*x*(b*x^2+a)^(7/2)-3/4096/b^3*a^5*x*(b*x^2+a)^(5/2)-15/16384/b^3*a^6*x*(b*x^2+
a)^(3/2)-45/32768*a^7*x*(b*x^2+a)^(1/2)/b^3-45/32768/b^(7/2)*a^8*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.53812, size = 656, normalized size = 3.25 \begin{align*} \left [\frac{315 \, a^{8} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (14336 \, b^{8} x^{15} + 66560 \, a b^{7} x^{13} + 119040 \, a^{2} b^{6} x^{11} + 98432 \, a^{3} b^{5} x^{9} + 32624 \, a^{4} b^{4} x^{7} + 168 \, a^{5} b^{3} x^{5} - 210 \, a^{6} b^{2} x^{3} + 315 \, a^{7} b x\right )} \sqrt{b x^{2} + a}}{458752 \, b^{4}}, \frac{315 \, a^{8} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (14336 \, b^{8} x^{15} + 66560 \, a b^{7} x^{13} + 119040 \, a^{2} b^{6} x^{11} + 98432 \, a^{3} b^{5} x^{9} + 32624 \, a^{4} b^{4} x^{7} + 168 \, a^{5} b^{3} x^{5} - 210 \, a^{6} b^{2} x^{3} + 315 \, a^{7} b x\right )} \sqrt{b x^{2} + a}}{229376 \, b^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

[1/458752*(315*a^8*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(14336*b^8*x^15 + 66560*a*b^7*x
^13 + 119040*a^2*b^6*x^11 + 98432*a^3*b^5*x^9 + 32624*a^4*b^4*x^7 + 168*a^5*b^3*x^5 - 210*a^6*b^2*x^3 + 315*a^
7*b*x)*sqrt(b*x^2 + a))/b^4, 1/229376*(315*a^8*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (14336*b^8*x^15 +
66560*a*b^7*x^13 + 119040*a^2*b^6*x^11 + 98432*a^3*b^5*x^9 + 32624*a^4*b^4*x^7 + 168*a^5*b^3*x^5 - 210*a^6*b^
2*x^3 + 315*a^7*b*x)*sqrt(b*x^2 + a))/b^4]

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Sympy [A]  time = 27.2185, size = 258, normalized size = 1.28 \begin{align*} \frac{45 a^{\frac{15}{2}} x}{32768 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 a^{\frac{13}{2}} x^{3}}{32768 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{11}{2}} x^{5}}{16384 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{4099 a^{\frac{9}{2}} x^{7}}{28672 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{8191 a^{\frac{7}{2}} b x^{9}}{14336 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{1699 a^{\frac{5}{2}} b^{2} x^{11}}{1792 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{725 a^{\frac{3}{2}} b^{3} x^{13}}{896 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{79 \sqrt{a} b^{4} x^{15}}{224 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{45 a^{8} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{32768 b^{\frac{7}{2}}} + \frac{b^{5} x^{17}}{16 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x**2+a)**(9/2),x)

[Out]

45*a**(15/2)*x/(32768*b**3*sqrt(1 + b*x**2/a)) + 15*a**(13/2)*x**3/(32768*b**2*sqrt(1 + b*x**2/a)) - 3*a**(11/
2)*x**5/(16384*b*sqrt(1 + b*x**2/a)) + 4099*a**(9/2)*x**7/(28672*sqrt(1 + b*x**2/a)) + 8191*a**(7/2)*b*x**9/(1
4336*sqrt(1 + b*x**2/a)) + 1699*a**(5/2)*b**2*x**11/(1792*sqrt(1 + b*x**2/a)) + 725*a**(3/2)*b**3*x**13/(896*s
qrt(1 + b*x**2/a)) + 79*sqrt(a)*b**4*x**15/(224*sqrt(1 + b*x**2/a)) - 45*a**8*asinh(sqrt(b)*x/sqrt(a))/(32768*
b**(7/2)) + b**5*x**17/(16*sqrt(a)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 2.54701, size = 180, normalized size = 0.89 \begin{align*} \frac{45 \, a^{8} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{32768 \, b^{\frac{7}{2}}} + \frac{1}{229376} \,{\left (\frac{315 \, a^{7}}{b^{3}} - 2 \,{\left (\frac{105 \, a^{6}}{b^{2}} - 4 \,{\left (\frac{21 \, a^{5}}{b} + 2 \,{\left (2039 \, a^{4} + 8 \,{\left (769 \, a^{3} b + 2 \,{\left (465 \, a^{2} b^{2} + 4 \,{\left (14 \, b^{4} x^{2} + 65 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

45/32768*a^8*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2) + 1/229376*(315*a^7/b^3 - 2*(105*a^6/b^2 - 4*(21*a
^5/b + 2*(2039*a^4 + 8*(769*a^3*b + 2*(465*a^2*b^2 + 4*(14*b^4*x^2 + 65*a*b^3)*x^2)*x^2)*x^2)*x^2)*x^2)*x^2)*s
qrt(b*x^2 + a)*x