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

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

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Rubi [A]  time = 0.17, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1804, 819, 778, 217, 206} \begin {gather*} -\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}} \end {gather*}

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 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])

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 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 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 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]

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*}

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Mathematica [A]  time = 0.41, size = 147, normalized size = 0.98 \begin {gather*} \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 {3 a^4 (16 B+35 C x)+14 a^3 b x^2 (12 B+25 C x)+14 a^2 b^2 x^4 (15 B+29 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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*(-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))/(a*b^4*(a + b*x^2)^(7/2)) + (Sqrt[a]*C*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/
Sqrt[a]])/(b^(9/2)*Sqrt[a + b*x^2])

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IntegrateAlgebraic [A]  time = 1.13, size = 138, normalized size = 0.92 \begin {gather*} \frac {-48 a^4 B-105 a^4 C x-168 a^3 b B x^2-350 a^3 b C x^3-210 a^2 b^2 B x^4-406 a^2 b^2 C x^5-105 a b^3 B x^6-176 a b^3 C x^7+15 A b^4 x^7}{105 a b^4 \left (a+b x^2\right )^{7/2}}-\frac {C \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-48*a^4*B - 105*a^4*C*x - 168*a^3*b*B*x^2 - 350*a^3*b*C*x^3 - 210*a^2*b^2*B*x^4 - 406*a^2*b^2*C*x^5 - 105*a*b
^3*B*x^6 + 15*A*b^4*x^7 - 176*a*b^3*C*x^7)/(105*a*b^4*(a + b*x^2)^(7/2)) - (C*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^
2]])/b^(9/2)

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fricas [A]  time = 1.11, size = 467, normalized size = 3.11 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.53, size = 138, normalized size = 0.92 \begin {gather*} -\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 {gather*}

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)

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maple [B]  time = 0.01, size = 277, normalized size = 1.85 \begin {gather*} -\frac {C \,x^{7}}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {B \,x^{6}}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {A \,x^{5}}{2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {2 B a \,x^{4}}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {C \,x^{5}}{5 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}-\frac {5 A a \,x^{3}}{8 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {8 B \,a^{2} x^{2}}{5 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}-\frac {15 A \,a^{2} x}{56 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}-\frac {C \,x^{3}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {3 A a x}{56 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{3}}-\frac {16 B \,a^{3}}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{4}}+\frac {A x}{14 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {A x}{7 \sqrt {b \,x^{2}+a}\, a \,b^{3}}-\frac {C x}{\sqrt {b \,x^{2}+a}\, b^{4}}+\frac {C \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {9}{2}}} \end {gather*}

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(b^(1/2)*x+(b*x^2+a)^(1/2))-B*x^6/b/(b*x^2+a)^(7/2)-2*B*a/b^2*x^4/(b*x^2+a)^(7/2)-8/5*B*a^2/b^3
*x^2/(b*x^2+a)^(7/2)-16/35*B*a^3/b^4/(b*x^2+a)^(7/2)-1/2*A*x^5/b/(b*x^2+a)^(7/2)-5/8*A*a/b^2*x^3/(b*x^2+a)^(7/
2)-15/56*A*a^2/b^3*x/(b*x^2+a)^(7/2)+3/56*A*a/b^3*x/(b*x^2+a)^(5/2)+1/14*A/b^3*x/(b*x^2+a)^(3/2)+1/7*A/a/b^3*x
/(b*x^2+a)^(1/2)

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maxima [B]  time = 1.63, size = 447, normalized size = 2.98 \begin {gather*} -\frac {1}{35} \, {\left (\frac {35 \, x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {70 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {56 \, a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {16 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}\right )} C x - \frac {B x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {C x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{15 \, b} - \frac {A x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {C x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b^{2}} - \frac {2 \, B a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {C a x^{3}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {5 \, A a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {8 \, B a^{2} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {139 \, C x}{105 \, \sqrt {b x^{2} + a} b^{4}} + \frac {17 \, C a x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {29 \, C a^{2} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} + \frac {A x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {A x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, A a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, A a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {C \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {9}{2}}} - \frac {16 \, B a^{3}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} \end {gather*}

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]

-1/35*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a*x^4/((b*x^2 + a)^(7/2)*b^2) + 56*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) +
16*a^3/((b*x^2 + a)^(7/2)*b^4))*C*x - B*x^6/((b*x^2 + a)^(7/2)*b) - 1/15*C*x*(15*x^4/((b*x^2 + a)^(5/2)*b) + 2
0*a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b - 1/2*A*x^5/((b*x^2 + a)^(7/2)*b) - 1/3*C*x
*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 - 2*B*a*x^4/((b*x^2 + a)^(7/2)*b^2) - C*a*x^3
/((b*x^2 + a)^(5/2)*b^3) - 5/8*A*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 8/5*B*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) + 139/1
05*C*x/(sqrt(b*x^2 + a)*b^4) + 17/105*C*a*x/((b*x^2 + a)^(3/2)*b^4) - 29/35*C*a^2*x/((b*x^2 + a)^(5/2)*b^4) +
1/14*A*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*A*x/(sqrt(b*x^2 + a)*a*b^3) + 3/56*A*a*x/((b*x^2 + a)^(5/2)*b^3) - 15/5
6*A*a^2*x/((b*x^2 + a)^(7/2)*b^3) + C*arcsinh(b*x/sqrt(a*b))/b^(9/2) - 16/35*B*a^3/((b*x^2 + a)^(7/2)*b^4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6\,\left (C\,x^2+B\,x+A\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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