3.47 \(\int \frac{x^7 (A+B x+C x^2)}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=213 \[ -\frac{x^5 (7 a B-x (A b-8 a C))}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (35 a B-6 x (A b-8 a C))}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x (35 a B-8 x (A b-8 a C))}{35 a b^4 \sqrt{a+b x^2}}-\frac{16 \sqrt{a+b x^2} (A b-8 a C)}{35 a b^5}-\frac{x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

[Out]

-(x^7*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (x^5*(7*a*B - (A*b - 8*a*C)*x))/(35*a*b^2*(a + b*x^2)
^(5/2)) - (x^3*(35*a*B - 6*(A*b - 8*a*C)*x))/(105*a*b^3*(a + b*x^2)^(3/2)) - (x*(35*a*B - 8*(A*b - 8*a*C)*x))/
(35*a*b^4*Sqrt[a + b*x^2]) - (16*(A*b - 8*a*C)*Sqrt[a + b*x^2])/(35*a*b^5) + (B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b
*x^2]])/b^(9/2)

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Rubi [A]  time = 0.324291, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1804, 819, 641, 217, 206} \[ -\frac{x^5 (7 a B-x (A b-8 a C))}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (35 a B-6 x (A b-8 a C))}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x (35 a B-8 x (A b-8 a C))}{35 a b^4 \sqrt{a+b x^2}}-\frac{16 \sqrt{a+b x^2} (A b-8 a C)}{35 a b^5}-\frac{x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

-(x^7*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (x^5*(7*a*B - (A*b - 8*a*C)*x))/(35*a*b^2*(a + b*x^2)
^(5/2)) - (x^3*(35*a*B - 6*(A*b - 8*a*C)*x))/(105*a*b^3*(a + b*x^2)^(3/2)) - (x*(35*a*B - 8*(A*b - 8*a*C)*x))/
(35*a*b^4*Sqrt[a + b*x^2]) - (16*(A*b - 8*a*C)*Sqrt[a + b*x^2])/(35*a*b^5) + (B*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 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[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^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac{x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x^6 (-7 a B+(A b-8 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac{x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{\int \frac{x^4 \left (-35 a^2 B+6 a (A b-8 a C) x\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2}\\ &=-\frac{x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{\int \frac{x^2 \left (-105 a^3 B+24 a^2 (A b-8 a C) x\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3 b^3}\\ &=-\frac{x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt{a+b x^2}}-\frac{\int \frac{-105 a^4 B+48 a^3 (A b-8 a C) x}{\sqrt{a+b x^2}} \, dx}{105 a^4 b^4}\\ &=-\frac{x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt{a+b x^2}}-\frac{16 (A b-8 a C) \sqrt{a+b x^2}}{35 a b^5}+\frac{B \int \frac{1}{\sqrt{a+b x^2}} \, dx}{b^4}\\ &=-\frac{x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt{a+b x^2}}-\frac{16 (A b-8 a C) \sqrt{a+b x^2}}{35 a b^5}+\frac{B \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{b^4}\\ &=-\frac{x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt{a+b x^2}}-\frac{16 (A b-8 a C) \sqrt{a+b x^2}}{35 a b^5}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.432563, size = 165, normalized size = 0.77 \[ \frac{14 a^2 b^2 x^2 (5 x (24 C x-5 B)-12 A)-3 a^3 b (16 A+7 x (5 B-64 C x))+384 a^4 C+14 a b^3 x^4 (x (60 C x-29 B)-15 A)+105 \sqrt{a} \sqrt{b} B \left (a+b x^2\right )^3 \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+b^4 x^6 (x (105 C x-176 B)-105 A)}{105 b^5 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

(384*a^4*C - 3*a^3*b*(16*A + 7*x*(5*B - 64*C*x)) + 14*a^2*b^2*x^2*(-12*A + 5*x*(-5*B + 24*C*x)) + 14*a*b^3*x^4
*(-15*A + x*(-29*B + 60*C*x)) + b^4*x^6*(-105*A + x*(-176*B + 105*C*x)) + 105*Sqrt[a]*Sqrt[b]*B*(a + b*x^2)^3*
Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(105*b^5*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.043, size = 265, normalized size = 1.2 \begin{align*}{\frac{C{x}^{8}}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+8\,{\frac{aC{x}^{6}}{{b}^{2} \left ( b{x}^{2}+a \right ) ^{7/2}}}+16\,{\frac{{a}^{2}C{x}^{4}}{{b}^{3} \left ( b{x}^{2}+a \right ) ^{7/2}}}+{\frac{64\,C{a}^{3}{x}^{2}}{5\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{128\,C{a}^{4}}{35\,{b}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{B{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{B{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{B{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Bx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{B\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}-{\frac{A{x}^{6}}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-2\,{\frac{aA{x}^{4}}{{b}^{2} \left ( b{x}^{2}+a \right ) ^{7/2}}}-{\frac{8\,A{a}^{2}{x}^{2}}{5\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{16\,A{a}^{3}}{35\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x)

[Out]

C*x^8/b/(b*x^2+a)^(7/2)+8*C/b^2*a*x^6/(b*x^2+a)^(7/2)+16*C/b^3*a^2*x^4/(b*x^2+a)^(7/2)+64/5*C/b^4*a^3*x^2/(b*x
^2+a)^(7/2)+128/35*C/b^5*a^4/(b*x^2+a)^(7/2)-1/7*B*x^7/b/(b*x^2+a)^(7/2)-1/5*B/b^2*x^5/(b*x^2+a)^(5/2)-1/3*B/b
^3*x^3/(b*x^2+a)^(3/2)-B/b^4*x/(b*x^2+a)^(1/2)+B/b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-A*x^6/b/(b*x^2+a)^(7/2)
-2*A/b^2*a*x^4/(b*x^2+a)^(7/2)-8/5*A/b^3*a^2*x^2/(b*x^2+a)^(7/2)-16/35*A/b^4*a^3/(b*x^2+a)^(7/2)

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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^7*(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 = 1.81342, size = 1160, normalized size = 5.45 \begin{align*} \left [\frac{105 \,{\left (B b^{4} x^{8} + 4 \, B a b^{3} x^{6} + 6 \, B a^{2} b^{2} x^{4} + 4 \, B a^{3} b x^{2} + B a^{4}\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \,{\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \,{\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \,{\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{210 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}, -\frac{105 \,{\left (B b^{4} x^{8} + 4 \, B a b^{3} x^{6} + 6 \, B a^{2} b^{2} x^{4} + 4 \, B a^{3} b x^{2} + B a^{4}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \,{\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \,{\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \,{\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

[1/210*(105*(B*b^4*x^8 + 4*B*a*b^3*x^6 + 6*B*a^2*b^2*x^4 + 4*B*a^3*b*x^2 + B*a^4)*sqrt(b)*log(-2*b*x^2 - 2*sqr
t(b*x^2 + a)*sqrt(b)*x - a) + 2*(105*C*b^4*x^8 - 176*B*b^4*x^7 - 406*B*a*b^3*x^5 - 350*B*a^2*b^2*x^3 + 105*(8*
C*a*b^3 - A*b^4)*x^6 - 105*B*a^3*b*x + 384*C*a^4 - 48*A*a^3*b + 210*(8*C*a^2*b^2 - A*a*b^3)*x^4 + 168*(8*C*a^3
*b - A*a^2*b^2)*x^2)*sqrt(b*x^2 + a))/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5), -1/10
5*(105*(B*b^4*x^8 + 4*B*a*b^3*x^6 + 6*B*a^2*b^2*x^4 + 4*B*a^3*b*x^2 + B*a^4)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b
*x^2 + a)) - (105*C*b^4*x^8 - 176*B*b^4*x^7 - 406*B*a*b^3*x^5 - 350*B*a^2*b^2*x^3 + 105*(8*C*a*b^3 - A*b^4)*x^
6 - 105*B*a^3*b*x + 384*C*a^4 - 48*A*a^3*b + 210*(8*C*a^2*b^2 - A*a*b^3)*x^4 + 168*(8*C*a^3*b - A*a^2*b^2)*x^2
)*sqrt(b*x^2 + a))/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*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**7*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.21738, size = 275, normalized size = 1.29 \begin{align*} \frac{{\left ({\left ({\left ({\left ({\left ({\left ({\left (\frac{105 \, C x}{b} - \frac{176 \, B}{b}\right )} x + \frac{105 \,{\left (8 \, C a^{4} b^{7} - A a^{3} b^{8}\right )}}{a^{3} b^{9}}\right )} x - \frac{406 \, B a}{b^{2}}\right )} x + \frac{210 \,{\left (8 \, C a^{5} b^{6} - A a^{4} b^{7}\right )}}{a^{3} b^{9}}\right )} x - \frac{350 \, B a^{2}}{b^{3}}\right )} x + \frac{168 \,{\left (8 \, C a^{6} b^{5} - A a^{5} b^{6}\right )}}{a^{3} b^{9}}\right )} x - \frac{105 \, B a^{3}}{b^{4}}\right )} x + \frac{48 \,{\left (8 \, C a^{7} b^{4} - A a^{6} b^{5}\right )}}{a^{3} b^{9}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{B \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^7*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

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