3.48 \(\int \frac{x^6 (A+B x+C x^2)}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

[Out]

-(x^6*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (x^4*(6*B + 7*C*x))/(35*b^2*(a + b*x^2)^(5/2)) - (x^2
*(24*B + 35*C*x))/(105*b^3*(a + b*x^2)^(3/2)) - (16*B + 35*C*x)/(35*b^4*Sqrt[a + b*x^2]) + (C*ArcTanh[(Sqrt[b]
*x)/Sqrt[a + b*x^2]])/b^(9/2)

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Rubi [A]  time = 0.165252, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1804, 819, 778, 217, 206} \[ -\frac{x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

-(x^6*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (x^4*(6*B + 7*C*x))/(35*b^2*(a + b*x^2)^(5/2)) - (x^2
*(24*B + 35*C*x))/(105*b^3*(a + b*x^2)^(3/2)) - (16*B + 35*C*x)/(35*b^4*Sqrt[a + b*x^2]) + (C*ArcTanh[(Sqrt[b]
*x)/Sqrt[a + b*x^2]])/b^(9/2)

Rule 1804

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[((c*x)^m*(a + b*x^2)^(p + 1)*(a*g - b*f*x))/(2*a*b*(p + 1)), x] + Dist[c/(2*a*b*(p + 1)), Int[
(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

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 \frac{x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x^5 (-6 a B-7 a C x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{\int \frac{x^3 \left (-24 a^2 B-35 a^2 C x\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{\int \frac{x \left (-48 a^3 B-105 a^3 C x\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3 b^3}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \int \frac{1}{\sqrt{a+b x^2}} \, dx}{b^4}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{b^4}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.268011, size = 147, normalized size = 0.98 \[ \frac{\sqrt{a} C \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a+b x^2}}-\frac{14 a^2 b^2 x^4 (15 B+29 C x)+14 a^3 b x^2 (12 B+25 C x)+3 a^4 (16 B+35 C x)+a b^3 x^6 (105 B+176 C x)-15 A b^4 x^7}{105 a b^4 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

-(-15*A*b^4*x^7 + 14*a^3*b*x^2*(12*B + 25*C*x) + 14*a^2*b^2*x^4*(15*B + 29*C*x) + 3*a^4*(16*B + 35*C*x) + a*b^
3*x^6*(105*B + 176*C*x))/(105*a*b^4*(a + b*x^2)^(7/2)) + (Sqrt[a]*C*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sq
rt[a]])/(b^(9/2)*Sqrt[a + b*x^2])

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Maple [B]  time = 0.012, size = 277, normalized size = 1.9 \begin{align*} -{\frac{C{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{C{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{C{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Cx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{C\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}-{\frac{B{x}^{6}}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-2\,{\frac{aB{x}^{4}}{{b}^{2} \left ( b{x}^{2}+a \right ) ^{7/2}}}-{\frac{8\,B{x}^{2}{a}^{2}}{5\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{16\,B{a}^{3}}{35\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{A{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,aA{x}^{3}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{15\,{a}^{2}Ax}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,aAx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Ax}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{Ax}{7\,{b}^{3}a}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*C*x^7/b/(b*x^2+a)^(7/2)-1/5*C/b^2*x^5/(b*x^2+a)^(5/2)-1/3*C/b^3*x^3/(b*x^2+a)^(3/2)-C/b^4*x/(b*x^2+a)^(1/
2)+C/b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-B*x^6/b/(b*x^2+a)^(7/2)-2*B/b^2*a*x^4/(b*x^2+a)^(7/2)-8/5*B/b^3*a^2
*x^2/(b*x^2+a)^(7/2)-16/35*B/b^4*a^3/(b*x^2+a)^(7/2)-1/2*A*x^5/b/(b*x^2+a)^(7/2)-5/8*A/b^2*a*x^3/(b*x^2+a)^(7/
2)-15/56*A/b^3*a^2*x/(b*x^2+a)^(7/2)+3/56*A/b^3*a*x/(b*x^2+a)^(5/2)+1/14*A/b^3*x/(b*x^2+a)^(3/2)+1/7*A/b^3/a*x
/(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*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74462, size = 1044, normalized size = 6.96 \begin{align*} \left [\frac{105 \,{\left (C a b^{4} x^{8} + 4 \, C a^{2} b^{3} x^{6} + 6 \, C a^{3} b^{2} x^{4} + 4 \, C a^{4} b x^{2} + C a^{5}\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (105 \, B a b^{4} x^{6} + 406 \, C a^{2} b^{3} x^{5} + 210 \, B a^{2} b^{3} x^{4} + 350 \, C a^{3} b^{2} x^{3} + 168 \, B a^{3} b^{2} x^{2} +{\left (176 \, C a b^{4} - 15 \, A b^{5}\right )} x^{7} + 105 \, C a^{4} b x + 48 \, B a^{4} b\right )} \sqrt{b x^{2} + a}}{210 \,{\left (a b^{9} x^{8} + 4 \, a^{2} b^{8} x^{6} + 6 \, a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}, -\frac{105 \,{\left (C a b^{4} x^{8} + 4 \, C a^{2} b^{3} x^{6} + 6 \, C a^{3} b^{2} x^{4} + 4 \, C a^{4} b x^{2} + C a^{5}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (105 \, B a b^{4} x^{6} + 406 \, C a^{2} b^{3} x^{5} + 210 \, B a^{2} b^{3} x^{4} + 350 \, C a^{3} b^{2} x^{3} + 168 \, B a^{3} b^{2} x^{2} +{\left (176 \, C a b^{4} - 15 \, A b^{5}\right )} x^{7} + 105 \, C a^{4} b x + 48 \, B a^{4} b\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a b^{9} x^{8} + 4 \, a^{2} b^{8} x^{6} + 6 \, a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(105*(C*a*b^4*x^8 + 4*C*a^2*b^3*x^6 + 6*C*a^3*b^2*x^4 + 4*C*a^4*b*x^2 + C*a^5)*sqrt(b)*log(-2*b*x^2 - 2
*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(105*B*a*b^4*x^6 + 406*C*a^2*b^3*x^5 + 210*B*a^2*b^3*x^4 + 350*C*a^3*b^2*x
^3 + 168*B*a^3*b^2*x^2 + (176*C*a*b^4 - 15*A*b^5)*x^7 + 105*C*a^4*b*x + 48*B*a^4*b)*sqrt(b*x^2 + a))/(a*b^9*x^
8 + 4*a^2*b^8*x^6 + 6*a^3*b^7*x^4 + 4*a^4*b^6*x^2 + a^5*b^5), -1/105*(105*(C*a*b^4*x^8 + 4*C*a^2*b^3*x^6 + 6*C
*a^3*b^2*x^4 + 4*C*a^4*b*x^2 + C*a^5)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (105*B*a*b^4*x^6 + 406*C*a
^2*b^3*x^5 + 210*B*a^2*b^3*x^4 + 350*C*a^3*b^2*x^3 + 168*B*a^3*b^2*x^2 + (176*C*a*b^4 - 15*A*b^5)*x^7 + 105*C*
a^4*b*x + 48*B*a^4*b)*sqrt(b*x^2 + a))/(a*b^9*x^8 + 4*a^2*b^8*x^6 + 6*a^3*b^7*x^4 + 4*a^4*b^6*x^2 + a^5*b^5)]

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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**6*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.22477, size = 186, normalized size = 1.24 \begin{align*} -\frac{{\left ({\left ({\left ({\left ({\left (x{\left (\frac{105 \, B}{b} + \frac{{\left (176 \, C a^{3} b^{7} - 15 \, A a^{2} b^{8}\right )} x}{a^{3} b^{8}}\right )} + \frac{406 \, C a}{b^{2}}\right )} x + \frac{210 \, B a}{b^{2}}\right )} x + \frac{350 \, C a^{2}}{b^{3}}\right )} x + \frac{168 \, B a^{2}}{b^{3}}\right )} x + \frac{105 \, C a^{3}}{b^{4}}\right )} x + \frac{48 \, B a^{3}}{b^{4}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{C \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*((((((x*(105*B/b + (176*C*a^3*b^7 - 15*A*a^2*b^8)*x/(a^3*b^8)) + 406*C*a/b^2)*x + 210*B*a/b^2)*x + 350*
C*a^2/b^3)*x + 168*B*a^2/b^3)*x + 105*C*a^3/b^4)*x + 48*B*a^3/b^4)/(b*x^2 + a)^(7/2) - C*log(abs(-sqrt(b)*x +
sqrt(b*x^2 + a)))/b^(9/2)