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

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

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Rubi [A]  time = 0.31, antiderivative size = 192, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1804, 1585, 1263, 1584, 452, 288, 217, 206} \begin {gather*} \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}} \end {gather*}

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 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 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 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 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 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 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^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 (\left (4 A b+\frac {3 a \left (b^2 B-a b C+a^2 D\right )}{b^2}\right ) x\right )-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*}

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Mathematica [A]  time = 0.46, size = 168, normalized size = 0.94 \begin {gather*} \frac {-105 a^6 D x-350 a^5 b D x^3-406 a^4 b^2 D x^5-176 a^3 b^3 D x^7+a^2 b^4 x^3 \left (35 A+21 B x^2+15 C x^4\right )+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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.65, size = 158, normalized size = 0.88 \begin {gather*} \frac {-105 a^6 D x-350 a^5 b D x^3-406 a^4 b^2 D x^5-176 a^3 b^3 D x^7+35 a^2 A b^4 x^3+21 a^2 b^4 B x^5+15 a^2 b^4 C x^7+28 a A b^5 x^5+6 a b^5 B x^7+8 A b^6 x^7}{105 a^3 b^4 \left (a+b x^2\right )^{7/2}}-\frac {D \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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 + 35*a^2*A*b^4*x^3 - 350*a^5*b*D*x^3 + 28*a*A*b^5*x^5 + 21*a^2*b^4*B*x^5 - 406*a^4*b^2*D*x^5 + 8
*A*b^6*x^7 + 6*a*b^5*B*x^7 + 15*a^2*b^4*C*x^7 - 176*a^3*b^3*D*x^7)/(105*a^3*b^4*(a + b*x^2)^(7/2)) - (D*Log[-(
Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(9/2)

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fricas [A]  time = 1.43, size = 491, normalized size = 2.74 \begin {gather*} \left [\frac {105 \, {\left (D a^{3} b^{4} x^{8} + 4 \, D a^{4} b^{3} x^{6} + 6 \, D a^{5} b^{2} x^{4} + 4 \, D a^{6} b x^{2} + D a^{7}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (105 \, D a^{6} b x + {\left (176 \, D a^{3} b^{4} - 15 \, C a^{2} b^{5} - 6 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 7 \, {\left (58 \, D a^{4} b^{3} - 3 \, B a^{2} b^{5} - 4 \, A a b^{6}\right )} x^{5} + 35 \, {\left (10 \, D a^{5} b^{2} - A a^{2} b^{5}\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{3} b^{9} x^{8} + 4 \, a^{4} b^{8} x^{6} + 6 \, a^{5} b^{7} x^{4} + 4 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}, -\frac {105 \, {\left (D a^{3} b^{4} x^{8} + 4 \, D a^{4} b^{3} x^{6} + 6 \, D a^{5} b^{2} x^{4} + 4 \, D a^{6} b x^{2} + D a^{7}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, D a^{6} b x + {\left (176 \, D a^{3} b^{4} - 15 \, C a^{2} b^{5} - 6 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 7 \, {\left (58 \, D a^{4} b^{3} - 3 \, B a^{2} b^{5} - 4 \, A a b^{6}\right )} x^{5} + 35 \, {\left (10 \, D a^{5} b^{2} - A a^{2} b^{5}\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{3} b^{9} x^{8} + 4 \, a^{4} b^{8} x^{6} + 6 \, a^{5} b^{7} x^{4} + 4 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}\right ] \end {gather*}

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]

[1/210*(105*(D*a^3*b^4*x^8 + 4*D*a^4*b^3*x^6 + 6*D*a^5*b^2*x^4 + 4*D*a^6*b*x^2 + D*a^7)*sqrt(b)*log(-2*b*x^2 -
2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(105*D*a^6*b*x + (176*D*a^3*b^4 - 15*C*a^2*b^5 - 6*B*a*b^6 - 8*A*b^7)*x^
7 + 7*(58*D*a^4*b^3 - 3*B*a^2*b^5 - 4*A*a*b^6)*x^5 + 35*(10*D*a^5*b^2 - A*a^2*b^5)*x^3)*sqrt(b*x^2 + a))/(a^3*
b^9*x^8 + 4*a^4*b^8*x^6 + 6*a^5*b^7*x^4 + 4*a^6*b^6*x^2 + a^7*b^5), -1/105*(105*(D*a^3*b^4*x^8 + 4*D*a^4*b^3*x
^6 + 6*D*a^5*b^2*x^4 + 4*D*a^6*b*x^2 + D*a^7)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (105*D*a^6*b*x + (
176*D*a^3*b^4 - 15*C*a^2*b^5 - 6*B*a*b^6 - 8*A*b^7)*x^7 + 7*(58*D*a^4*b^3 - 3*B*a^2*b^5 - 4*A*a*b^6)*x^5 + 35*
(10*D*a^5*b^2 - A*a^2*b^5)*x^3)*sqrt(b*x^2 + a))/(a^3*b^9*x^8 + 4*a^4*b^8*x^6 + 6*a^5*b^7*x^4 + 4*a^6*b^6*x^2
+ a^7*b^5)]

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giac [A]  time = 0.55, size = 160, normalized size = 0.89 \begin {gather*} -\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 {gather*}

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)

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maple [B]  time = 0.01, size = 363, normalized size = 2.03 \begin {gather*} -\frac {D x^{7}}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {C \,x^{5}}{2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {D x^{5}}{5 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}-\frac {B \,x^{3}}{4 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {5 C a \,x^{3}}{8 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {A x}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {3 B a x}{28 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {15 C \,a^{2} x}{56 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}-\frac {D x^{3}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {A x}{35 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a b}+\frac {3 B x}{140 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}+\frac {3 C a x}{56 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{3}}+\frac {4 A x}{105 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b}+\frac {B x}{35 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,b^{2}}+\frac {C x}{14 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {8 A x}{105 \sqrt {b \,x^{2}+a}\, a^{3} b}+\frac {2 B x}{35 \sqrt {b \,x^{2}+a}\, a^{2} b^{2}}+\frac {C x}{7 \sqrt {b \,x^{2}+a}\, a \,b^{3}}-\frac {D x}{\sqrt {b \,x^{2}+a}\, b^{4}}+\frac {D \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^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(b^(1/2)*x+(b*x^2+a)^(1/2))-1/2*C*x^5/b/(b*x^2+a)^(7/2)-5/8*C*a/b^2*x^3/(b*x^2+a)^(7/2)-15/56*C
*a^2/b^3*x/(b*x^2+a)^(7/2)+3/56*C*a/b^3*x/(b*x^2+a)^(5/2)+1/14*C/b^3*x/(b*x^2+a)^(3/2)+1/7*C/a/b^3*x/(b*x^2+a)
^(1/2)-1/4*B*x^3/b/(b*x^2+a)^(7/2)-3/28*B*a/b^2*x/(b*x^2+a)^(7/2)+3/140*B/b^2*x/(b*x^2+a)^(5/2)+1/35*B/a/b^2*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/a/b*x/(b*x^2+a)^(5/2)+4/105
*A/a^2/b*x/(b*x^2+a)^(3/2)+8/105*A/a^3/b*x/(b*x^2+a)^(1/2)

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maxima [B]  time = 1.66, size = 533, 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 )} D x - \frac {D 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 {C x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {D 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 {D a x^{3}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {5 \, C a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {B x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {139 \, D x}{105 \, \sqrt {b x^{2} + a} b^{4}} + \frac {17 \, D a x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {29 \, D a^{2} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} + \frac {C x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {C x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, C a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, C a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {3 \, B x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, B x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, B a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, A x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, A x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} + \frac {D \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {9}{2}}} \end {gather*}

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]

-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))*D*x - 1/15*D*x*(15*x^4/((b*x^2 + a)^(5/2)*b) + 20*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*C*x^5/((b*x^2 + a)^(7/2)*b) - 1/3*D*x*(3*x^2/((b*x^2 + a)^(3/2)*b)
+ 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 - D*a*x^3/((b*x^2 + a)^(5/2)*b^3) - 5/8*C*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 1
/4*B*x^3/((b*x^2 + a)^(7/2)*b) + 139/105*D*x/(sqrt(b*x^2 + a)*b^4) + 17/105*D*a*x/((b*x^2 + a)^(3/2)*b^4) - 29
/35*D*a^2*x/((b*x^2 + a)^(5/2)*b^4) + 1/14*C*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*C*x/(sqrt(b*x^2 + a)*a*b^3) + 3/5
6*C*a*x/((b*x^2 + a)^(5/2)*b^3) - 15/56*C*a^2*x/((b*x^2 + a)^(7/2)*b^3) + 3/140*B*x/((b*x^2 + a)^(5/2)*b^2) +
2/35*B*x/(sqrt(b*x^2 + a)*a^2*b^2) + 1/35*B*x/((b*x^2 + a)^(3/2)*a*b^2) - 3/28*B*a*x/((b*x^2 + a)^(7/2)*b^2) -
1/7*A*x/((b*x^2 + a)^(7/2)*b) + 8/105*A*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*A*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/3
5*A*x/((b*x^2 + a)^(5/2)*a*b) + D*arcsinh(b*x/sqrt(a*b))/b^(9/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\left (A+B\,x^2+C\,x^4+x^6\,D\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^2*(A + B*x^2 + C*x^4 + x^6*D))/(a + b*x^2)^(9/2),x)

[Out]

int((x^2*(A + B*x^2 + C*x^4 + x^6*D))/(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**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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