3.162 \(\int \frac{x^2 (A+B x^2+C x^4+D x^6)}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=179 \[ \frac{x^7 \left (15 a^2 b C-176 a^3 D+6 a b^2 B+8 A b^3\right )}{105 a^3 b \left (a+b x^2\right )^{7/2}}+\frac{x^5 \left (-58 a^3 D+3 a b^2 B+4 A b^3\right )}{15 a^2 b^2 \left (a+b x^2\right )^{7/2}}+\frac{x^3 \left (A b^3-10 a^3 D\right )}{3 a b^3 \left (a+b x^2\right )^{7/2}}-\frac{a^3 D x}{b^4 \left (a+b x^2\right )^{7/2}}+\frac{D \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

[Out]

-((a^3*D*x)/(b^4*(a + b*x^2)^(7/2))) + ((A*b^3 - 10*a^3*D)*x^3)/(3*a*b^3*(a + b*x^2)^(7/2)) + ((4*A*b^3 + 3*a*
b^2*B - 58*a^3*D)*x^5)/(15*a^2*b^2*(a + b*x^2)^(7/2)) + ((8*A*b^3 + 6*a*b^2*B + 15*a^2*b*C - 176*a^3*D)*x^7)/(
105*a^3*b*(a + b*x^2)^(7/2)) + (D*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(9/2)

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Rubi [A]  time = 0.313859, antiderivative size = 192, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1804, 1585, 1263, 1584, 452, 288, 217, 206} \[ \frac{x^3 \left (a \left (-71 a^2 D+15 a b C+6 b^2 B\right )+8 A b^3\right )}{105 a^3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{x^3 \left (a \left (17 a^2 D-10 a b C+3 b^2 B\right )+4 A b^3\right )}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac{x^3 \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac{D x}{b^4 \sqrt{a+b x^2}}+\frac{D \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]

[Out]

((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^3)/(7*a*(a + b*x^2)^(7/2)) + ((4*A*b^3 + a*(3*b^2*B - 10*a*b*C + 17*a
^2*D))*x^3)/(35*a^2*b^3*(a + b*x^2)^(5/2)) + ((8*A*b^3 + a*(6*b^2*B + 15*a*b*C - 71*a^2*D))*x^3)/(105*a^3*b^3*
(a + b*x^2)^(3/2)) - (D*x)/(b^4*Sqrt[a + b*x^2]) + (D*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 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 1263

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*
x^4)^p, d + e*x^2, x], x, 0]}, -Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(2*d*f*(q + 1)), x] + Dist[f/(2*d*(
q + 1)), Int[(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*x*Qx + R*(m + 2*q + 3)*x, x], x], x]] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[q, -1] && GtQ[m, 0]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 452

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

Rule 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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 \frac{x^2 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^3}{7 a \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x \left (-\left (4 A b+\frac{3 a \left (b^2 B-a b C+a^2 D\right )}{b^2}\right ) x-7 a \left (C-\frac{a D}{b}\right ) x^3-7 a D x^5\right )}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^3}{7 a \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x^2 \left (-4 A b-\frac{3 a \left (b^2 B-a b C+a^2 D\right )}{b^2}-7 a \left (C-\frac{a D}{b}\right ) x^2-7 a D x^4\right )}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^3}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (4 A b^3+a \left (3 b^2 B-10 a b C+17 a^2 D\right )\right ) x^3}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac{\int \frac{x \left (\left (8 A b+\frac{3 a \left (2 b^2 B+5 a b C-12 a^2 D\right )}{b^2}\right ) x+\frac{35 a^2 D x^3}{b}\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^3}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (4 A b^3+a \left (3 b^2 B-10 a b C+17 a^2 D\right )\right ) x^3}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac{\int \frac{x^2 \left (8 A b+\frac{3 a \left (2 b^2 B+5 a b C-12 a^2 D\right )}{b^2}+\frac{35 a^2 D x^2}{b}\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^3}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (4 A b^3+a \left (3 b^2 B-10 a b C+17 a^2 D\right )\right ) x^3}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac{\left (8 A b^3+a \left (6 b^2 B+15 a b C-71 a^2 D\right )\right ) x^3}{105 a^3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{D \int \frac{x^2}{\left (a+b x^2\right )^{3/2}} \, dx}{b^3}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^3}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (4 A b^3+a \left (3 b^2 B-10 a b C+17 a^2 D\right )\right ) x^3}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac{\left (8 A b^3+a \left (6 b^2 B+15 a b C-71 a^2 D\right )\right ) x^3}{105 a^3 b^3 \left (a+b x^2\right )^{3/2}}-\frac{D x}{b^4 \sqrt{a+b x^2}}+\frac{D \int \frac{1}{\sqrt{a+b x^2}} \, dx}{b^4}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^3}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (4 A b^3+a \left (3 b^2 B-10 a b C+17 a^2 D\right )\right ) x^3}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac{\left (8 A b^3+a \left (6 b^2 B+15 a b C-71 a^2 D\right )\right ) x^3}{105 a^3 b^3 \left (a+b x^2\right )^{3/2}}-\frac{D x}{b^4 \sqrt{a+b x^2}}+\frac{D \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{b^4}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^3}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (4 A b^3+a \left (3 b^2 B-10 a b C+17 a^2 D\right )\right ) x^3}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac{\left (8 A b^3+a \left (6 b^2 B+15 a b C-71 a^2 D\right )\right ) x^3}{105 a^3 b^3 \left (a+b x^2\right )^{3/2}}-\frac{D x}{b^4 \sqrt{a+b x^2}}+\frac{D \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.410325, size = 168, normalized size = 0.94 \[ \frac{a^2 b^4 x^3 \left (35 A+21 B x^2+15 C x^4\right )-176 a^3 b^3 D x^7-406 a^4 b^2 D x^5-350 a^5 b D x^3-105 a^6 D x+2 a b^5 x^5 \left (14 A+3 B x^2\right )+8 A b^6 x^7}{105 a^3 b^4 \left (a+b x^2\right )^{7/2}}+\frac{\sqrt{a} D \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]

[Out]

(-105*a^6*D*x - 350*a^5*b*D*x^3 - 406*a^4*b^2*D*x^5 + 8*A*b^6*x^7 - 176*a^3*b^3*D*x^7 + 2*a*b^5*x^5*(14*A + 3*
B*x^2) + a^2*b^4*x^3*(35*A + 21*B*x^2 + 15*C*x^4))/(105*a^3*b^4*(a + b*x^2)^(7/2)) + (Sqrt[a]*D*Sqrt[1 + (b*x^
2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(b^(9/2)*Sqrt[a + b*x^2])

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Maple [B]  time = 0.008, size = 363, normalized size = 2. \begin{align*} -{\frac{D{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{D{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{D{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Dx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{D\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}-{\frac{C{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,aC{x}^{3}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{15\,{a}^{2}Cx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,aCx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Cx}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{Cx}{7\,{b}^{3}a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{B{x}^{3}}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{3\,Bax}{28\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,Bx}{140\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Bx}{35\,{b}^{2}a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Bx}{35\,{b}^{2}{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{Ax}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{Ax}{35\,ab} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{4\,Ax}{105\,b{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,Ax}{105\,b{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x)

[Out]

-1/7*D*x^7/b/(b*x^2+a)^(7/2)-1/5*D/b^2*x^5/(b*x^2+a)^(5/2)-1/3*D/b^3*x^3/(b*x^2+a)^(3/2)-D*x/b^4/(b*x^2+a)^(1/
2)+D/b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-1/2*C*x^5/b/(b*x^2+a)^(7/2)-5/8*C/b^2*a*x^3/(b*x^2+a)^(7/2)-15/56*C
/b^3*a^2*x/(b*x^2+a)^(7/2)+3/56*C/b^3*a*x/(b*x^2+a)^(5/2)+1/14*C/b^3*x/(b*x^2+a)^(3/2)+1/7*C/b^3/a*x/(b*x^2+a)
^(1/2)-1/4*B*x^3/b/(b*x^2+a)^(7/2)-3/28*B/b^2*a*x/(b*x^2+a)^(7/2)+3/140*B/b^2*x/(b*x^2+a)^(5/2)+1/35*B/b^2/a*x
/(b*x^2+a)^(3/2)+2/35*B*x/a^2/b^2/(b*x^2+a)^(1/2)-1/7*A/b*x/(b*x^2+a)^(7/2)+1/35*A/b/a*x/(b*x^2+a)^(5/2)+4/105
*A/b/a^2*x/(b*x^2+a)^(3/2)+8/105*A/b/a^3*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^2*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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**2*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.34302, size = 216, normalized size = 1.21 \begin{align*} -\frac{{\left ({\left (x^{2}{\left (\frac{{\left (176 \, D a^{3} b^{6} - 15 \, C a^{2} b^{7} - 6 \, B a b^{8} - 8 \, A b^{9}\right )} x^{2}}{a^{3} b^{7}} + \frac{7 \,{\left (58 \, D a^{4} b^{5} - 3 \, B a^{2} b^{7} - 4 \, A a b^{8}\right )}}{a^{3} b^{7}}\right )} + \frac{35 \,{\left (10 \, D a^{5} b^{4} - A a^{2} b^{7}\right )}}{a^{3} b^{7}}\right )} x^{2} + \frac{105 \, D a^{3}}{b^{4}}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{D \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^2*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

-1/105*((x^2*((176*D*a^3*b^6 - 15*C*a^2*b^7 - 6*B*a*b^8 - 8*A*b^9)*x^2/(a^3*b^7) + 7*(58*D*a^4*b^5 - 3*B*a^2*b
^7 - 4*A*a*b^8)/(a^3*b^7)) + 35*(10*D*a^5*b^4 - A*a^2*b^7)/(a^3*b^7))*x^2 + 105*D*a^3/b^4)*x/(b*x^2 + a)^(7/2)
- D*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)