3.1.49 \(\int \frac {x^5 (A+B x+C x^2)}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=132 \[ -\frac {4 (6 a C+A b)}{35 a b^4 \sqrt {a+b x^2}}+\frac {4 (6 a C+A b)}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac {x^4 (6 a C+A b-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]

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Rubi [A]  time = 0.17, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1804, 805, 266, 43} \begin {gather*} -\frac {x^4 (6 a C+A b-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {4 (6 a C+A b)}{35 a b^4 \sqrt {a+b x^2}}+\frac {4 (6 a C+A b)}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac {x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 805

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

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

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Mathematica [A]  time = 0.11, size = 89, normalized size = 0.67 \begin {gather*} \frac {-48 a^4 C-8 a^3 b \left (A+21 C x^2\right )-14 a^2 b^2 x^2 \left (2 A+15 C x^2\right )-35 a b^3 x^4 \left (A+3 C x^2\right )+15 b^4 B x^7}{105 a b^4 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

(-48*a^4*C + 15*b^4*B*x^7 - 35*a*b^3*x^4*(A + 3*C*x^2) - 14*a^2*b^2*x^2*(2*A + 15*C*x^2) - 8*a^3*b*(A + 21*C*x
^2))/(105*a*b^4*(a + b*x^2)^(7/2))

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IntegrateAlgebraic [A]  time = 1.21, size = 98, normalized size = 0.74 \begin {gather*} \frac {-48 a^4 C-8 a^3 A b-168 a^3 b C x^2-28 a^2 A b^2 x^2-210 a^2 b^2 C x^4-35 a A b^3 x^4-105 a b^3 C x^6+15 b^4 B x^7}{105 a b^4 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^5*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

(-8*a^3*A*b - 48*a^4*C - 28*a^2*A*b^2*x^2 - 168*a^3*b*C*x^2 - 35*a*A*b^3*x^4 - 210*a^2*b^2*C*x^4 - 105*a*b^3*C
*x^6 + 15*b^4*B*x^7)/(105*a*b^4*(a + b*x^2)^(7/2))

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fricas [A]  time = 0.68, size = 137, normalized size = 1.04 \begin {gather*} \frac {{\left (15 \, B b^{4} x^{7} - 105 \, C a b^{3} x^{6} - 48 \, C a^{4} - 8 \, A a^{3} b - 35 \, {\left (6 \, C a^{2} b^{2} + A a b^{3}\right )} x^{4} - 28 \, {\left (6 \, C a^{3} b + A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a b^{8} x^{8} + 4 \, a^{2} b^{7} x^{6} + 6 \, a^{3} b^{6} x^{4} + 4 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/105*(15*B*b^4*x^7 - 105*C*a*b^3*x^6 - 48*C*a^4 - 8*A*a^3*b - 35*(6*C*a^2*b^2 + A*a*b^3)*x^4 - 28*(6*C*a^3*b
+ A*a^2*b^2)*x^2)*sqrt(b*x^2 + a)/(a*b^8*x^8 + 4*a^2*b^7*x^6 + 6*a^3*b^6*x^4 + 4*a^4*b^5*x^2 + a^5*b^4)

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giac [A]  time = 0.62, size = 112, normalized size = 0.85 \begin {gather*} \frac {{\left (5 \, {\left (3 \, {\left (\frac {B x}{a} - \frac {7 \, C}{b}\right )} x^{2} - \frac {7 \, {\left (6 \, C a^{4} b^{2} + A a^{3} b^{3}\right )}}{a^{3} b^{4}}\right )} x^{2} - \frac {28 \, {\left (6 \, C a^{5} b + A a^{4} b^{2}\right )}}{a^{3} b^{4}}\right )} x^{2} - \frac {8 \, {\left (6 \, C a^{6} + A a^{5} b\right )}}{a^{3} b^{4}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/105*((5*(3*(B*x/a - 7*C/b)*x^2 - 7*(6*C*a^4*b^2 + A*a^3*b^3)/(a^3*b^4))*x^2 - 28*(6*C*a^5*b + A*a^4*b^2)/(a^
3*b^4))*x^2 - 8*(6*C*a^6 + A*a^5*b)/(a^3*b^4))/(b*x^2 + a)^(7/2)

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maple [A]  time = 0.00, size = 95, normalized size = 0.72 \begin {gather*} -\frac {-15 B \,x^{7} b^{4}+105 C \,x^{6} a \,b^{3}+35 A a \,b^{3} x^{4}+210 C \,a^{2} b^{2} x^{4}+28 A \,a^{2} b^{2} x^{2}+168 C \,a^{3} b \,x^{2}+8 A \,a^{3} b +48 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x)

[Out]

-1/105*(-15*B*b^4*x^7+105*C*a*b^3*x^6+35*A*a*b^3*x^4+210*C*a^2*b^2*x^4+28*A*a^2*b^2*x^2+168*C*a^3*b*x^2+8*A*a^
3*b+48*C*a^4)/(b*x^2+a)^(7/2)/a/b^4

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maxima [B]  time = 1.44, size = 240, normalized size = 1.82 \begin {gather*} -\frac {C x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {2 \, C a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x^{4}}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {5 \, B a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {8 \, C a^{2} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {4 \, A a x^{2}}{15 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {B x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {B x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, B a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, B a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {16 \, C a^{3}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} - \frac {8 \, A a^{2}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-C*x^6/((b*x^2 + a)^(7/2)*b) - 1/2*B*x^5/((b*x^2 + a)^(7/2)*b) - 2*C*a*x^4/((b*x^2 + a)^(7/2)*b^2) - 1/3*A*x^4
/((b*x^2 + a)^(7/2)*b) - 5/8*B*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 8/5*C*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) - 4/15*A*
a*x^2/((b*x^2 + a)^(7/2)*b^2) + 1/14*B*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*B*x/(sqrt(b*x^2 + a)*a*b^3) + 3/56*B*a*
x/((b*x^2 + a)^(5/2)*b^3) - 15/56*B*a^2*x/((b*x^2 + a)^(7/2)*b^3) - 16/35*C*a^3/((b*x^2 + a)^(7/2)*b^4) - 8/10
5*A*a^2/((b*x^2 + a)^(7/2)*b^3)

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mupad [B]  time = 1.27, size = 196, normalized size = 1.48 \begin {gather*} \frac {\frac {a\,\left (\frac {C}{3\,b^3}-\frac {7\,A\,b-14\,C\,a}{21\,a\,b^3}\right )}{b}-\frac {3\,B\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {\frac {a^2\,\left (\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}\right )}{b^2}+\frac {B\,a^2\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {C}{b^4}-\frac {B\,x}{7\,a\,b^3}}{\sqrt {b\,x^2+a}}-\frac {\frac {a\,\left (\frac {7\,C\,a^2-7\,A\,a\,b}{35\,a\,b^3}+\frac {a\,\left (\frac {C}{5\,b^2}-\frac {7\,A\,b^2-7\,C\,a\,b}{35\,a\,b^3}\right )}{b}\right )}{b}-\frac {3\,B\,a\,x}{7\,b^3}}{{\left (b\,x^2+a\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x)

[Out]

((a*(C/(3*b^3) - (7*A*b - 14*C*a)/(21*a*b^3)))/b - (3*B*x)/(7*b^3))/(a + b*x^2)^(3/2) - ((a^2*(A/(7*b) - (C*a)
/(7*b^2)))/b^2 + (B*a^2*x)/(7*b^3))/(a + b*x^2)^(7/2) - (C/b^4 - (B*x)/(7*a*b^3))/(a + b*x^2)^(1/2) - ((a*((7*
C*a^2 - 7*A*a*b)/(35*a*b^3) + (a*(C/(5*b^2) - (7*A*b^2 - 7*C*a*b)/(35*a*b^3)))/b))/b - (3*B*a*x)/(7*b^3))/(a +
b*x^2)^(5/2)

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

[Out]

Timed out

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