3.7.9 \(\int x (a+b x^2)^2 (c+d x^2)^{5/2} \, dx\)

Optimal. Leaf size=77 \[ -\frac {2 b \left (c+d x^2\right )^{9/2} (b c-a d)}{9 d^3}+\frac {\left (c+d x^2\right )^{7/2} (b c-a d)^2}{7 d^3}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^3} \]

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Rubi [A]  time = 0.06, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {444, 43} \begin {gather*} -\frac {2 b \left (c+d x^2\right )^{9/2} (b c-a d)}{9 d^3}+\frac {\left (c+d x^2\right )^{7/2} (b c-a d)^2}{7 d^3}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]

[Out]

((b*c - a*d)^2*(c + d*x^2)^(7/2))/(7*d^3) - (2*b*(b*c - a*d)*(c + d*x^2)^(9/2))/(9*d^3) + (b^2*(c + d*x^2)^(11
/2))/(11*d^3)

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 444

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

Rubi steps

\begin {align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (a+b x)^2 (c+d x)^{5/2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {(-b c+a d)^2 (c+d x)^{5/2}}{d^2}-\frac {2 b (b c-a d) (c+d x)^{7/2}}{d^2}+\frac {b^2 (c+d x)^{9/2}}{d^2}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d)^2 \left (c+d x^2\right )^{7/2}}{7 d^3}-\frac {2 b (b c-a d) \left (c+d x^2\right )^{9/2}}{9 d^3}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 67, normalized size = 0.87 \begin {gather*} \frac {\left (c+d x^2\right )^{7/2} \left (99 a^2 d^2+22 a b d \left (7 d x^2-2 c\right )+b^2 \left (8 c^2-28 c d x^2+63 d^2 x^4\right )\right )}{693 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]

[Out]

((c + d*x^2)^(7/2)*(99*a^2*d^2 + 22*a*b*d*(-2*c + 7*d*x^2) + b^2*(8*c^2 - 28*c*d*x^2 + 63*d^2*x^4)))/(693*d^3)

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IntegrateAlgebraic [A]  time = 0.05, size = 72, normalized size = 0.94 \begin {gather*} \frac {\left (c+d x^2\right )^{7/2} \left (99 a^2 d^2-44 a b c d+154 a b d^2 x^2+8 b^2 c^2-28 b^2 c d x^2+63 b^2 d^2 x^4\right )}{693 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]

[Out]

((c + d*x^2)^(7/2)*(8*b^2*c^2 - 44*a*b*c*d + 99*a^2*d^2 - 28*b^2*c*d*x^2 + 154*a*b*d^2*x^2 + 63*b^2*d^2*x^4))/
(693*d^3)

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fricas [B]  time = 1.46, size = 178, normalized size = 2.31 \begin {gather*} \frac {{\left (63 \, b^{2} d^{5} x^{10} + 7 \, {\left (23 \, b^{2} c d^{4} + 22 \, a b d^{5}\right )} x^{8} + 8 \, b^{2} c^{5} - 44 \, a b c^{4} d + 99 \, a^{2} c^{3} d^{2} + {\left (113 \, b^{2} c^{2} d^{3} + 418 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{6} + 3 \, {\left (b^{2} c^{3} d^{2} + 110 \, a b c^{2} d^{3} + 99 \, a^{2} c d^{4}\right )} x^{4} - {\left (4 \, b^{2} c^{4} d - 22 \, a b c^{3} d^{2} - 297 \, a^{2} c^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{693 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^2*(d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

1/693*(63*b^2*d^5*x^10 + 7*(23*b^2*c*d^4 + 22*a*b*d^5)*x^8 + 8*b^2*c^5 - 44*a*b*c^4*d + 99*a^2*c^3*d^2 + (113*
b^2*c^2*d^3 + 418*a*b*c*d^4 + 99*a^2*d^5)*x^6 + 3*(b^2*c^3*d^2 + 110*a*b*c^2*d^3 + 99*a^2*c*d^4)*x^4 - (4*b^2*
c^4*d - 22*a*b*c^3*d^2 - 297*a^2*c^2*d^3)*x^2)*sqrt(d*x^2 + c)/d^3

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giac [A]  time = 0.32, size = 98, normalized size = 1.27 \begin {gather*} \frac {63 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} b^{2} - 154 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} c + 99 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} + 154 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a b d - 198 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c d + 99 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{2}}{693 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^2*(d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

1/693*(63*(d*x^2 + c)^(11/2)*b^2 - 154*(d*x^2 + c)^(9/2)*b^2*c + 99*(d*x^2 + c)^(7/2)*b^2*c^2 + 154*(d*x^2 + c
)^(9/2)*a*b*d - 198*(d*x^2 + c)^(7/2)*a*b*c*d + 99*(d*x^2 + c)^(7/2)*a^2*d^2)/d^3

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maple [A]  time = 0.01, size = 69, normalized size = 0.90 \begin {gather*} \frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} \left (63 b^{2} x^{4} d^{2}+154 a b \,d^{2} x^{2}-28 b^{2} c d \,x^{2}+99 a^{2} d^{2}-44 a b c d +8 b^{2} c^{2}\right )}{693 d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^2*(d*x^2+c)^(5/2),x)

[Out]

1/693*(d*x^2+c)^(7/2)*(63*b^2*d^2*x^4+154*a*b*d^2*x^2-28*b^2*c*d*x^2+99*a^2*d^2-44*a*b*c*d+8*b^2*c^2)/d^3

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maxima [A]  time = 0.89, size = 115, normalized size = 1.49 \begin {gather*} \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} x^{4}}{11 \, d} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c x^{2}}{99 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b x^{2}}{9 \, d} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2}}{693 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c}{63 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{7 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^2*(d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

1/11*(d*x^2 + c)^(7/2)*b^2*x^4/d - 4/99*(d*x^2 + c)^(7/2)*b^2*c*x^2/d^2 + 2/9*(d*x^2 + c)^(7/2)*a*b*x^2/d + 8/
693*(d*x^2 + c)^(7/2)*b^2*c^2/d^3 - 4/63*(d*x^2 + c)^(7/2)*a*b*c/d^2 + 1/7*(d*x^2 + c)^(7/2)*a^2/d

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mupad [B]  time = 0.80, size = 98, normalized size = 1.27 \begin {gather*} \frac {d\,\left (\frac {2\,a\,b\,{\left (d\,x^2+c\right )}^{9/2}}{9}-\frac {2\,a\,b\,c\,{\left (d\,x^2+c\right )}^{7/2}}{7}\right )+\frac {b^2\,{\left (d\,x^2+c\right )}^{11/2}}{11}-\frac {2\,b^2\,c\,{\left (d\,x^2+c\right )}^{9/2}}{9}+\frac {a^2\,d^2\,{\left (d\,x^2+c\right )}^{7/2}}{7}+\frac {b^2\,c^2\,{\left (d\,x^2+c\right )}^{7/2}}{7}}{d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^2*(c + d*x^2)^(5/2),x)

[Out]

(d*((2*a*b*(c + d*x^2)^(9/2))/9 - (2*a*b*c*(c + d*x^2)^(7/2))/7) + (b^2*(c + d*x^2)^(11/2))/11 - (2*b^2*c*(c +
d*x^2)^(9/2))/9 + (a^2*d^2*(c + d*x^2)^(7/2))/7 + (b^2*c^2*(c + d*x^2)^(7/2))/7)/d^3

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sympy [A]  time = 14.55, size = 384, normalized size = 4.99 \begin {gather*} \begin {cases} \frac {a^{2} c^{3} \sqrt {c + d x^{2}}}{7 d} + \frac {3 a^{2} c^{2} x^{2} \sqrt {c + d x^{2}}}{7} + \frac {3 a^{2} c d x^{4} \sqrt {c + d x^{2}}}{7} + \frac {a^{2} d^{2} x^{6} \sqrt {c + d x^{2}}}{7} - \frac {4 a b c^{4} \sqrt {c + d x^{2}}}{63 d^{2}} + \frac {2 a b c^{3} x^{2} \sqrt {c + d x^{2}}}{63 d} + \frac {10 a b c^{2} x^{4} \sqrt {c + d x^{2}}}{21} + \frac {38 a b c d x^{6} \sqrt {c + d x^{2}}}{63} + \frac {2 a b d^{2} x^{8} \sqrt {c + d x^{2}}}{9} + \frac {8 b^{2} c^{5} \sqrt {c + d x^{2}}}{693 d^{3}} - \frac {4 b^{2} c^{4} x^{2} \sqrt {c + d x^{2}}}{693 d^{2}} + \frac {b^{2} c^{3} x^{4} \sqrt {c + d x^{2}}}{231 d} + \frac {113 b^{2} c^{2} x^{6} \sqrt {c + d x^{2}}}{693} + \frac {23 b^{2} c d x^{8} \sqrt {c + d x^{2}}}{99} + \frac {b^{2} d^{2} x^{10} \sqrt {c + d x^{2}}}{11} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a)**2*(d*x**2+c)**(5/2),x)

[Out]

Piecewise((a**2*c**3*sqrt(c + d*x**2)/(7*d) + 3*a**2*c**2*x**2*sqrt(c + d*x**2)/7 + 3*a**2*c*d*x**4*sqrt(c + d
*x**2)/7 + a**2*d**2*x**6*sqrt(c + d*x**2)/7 - 4*a*b*c**4*sqrt(c + d*x**2)/(63*d**2) + 2*a*b*c**3*x**2*sqrt(c
+ d*x**2)/(63*d) + 10*a*b*c**2*x**4*sqrt(c + d*x**2)/21 + 38*a*b*c*d*x**6*sqrt(c + d*x**2)/63 + 2*a*b*d**2*x**
8*sqrt(c + d*x**2)/9 + 8*b**2*c**5*sqrt(c + d*x**2)/(693*d**3) - 4*b**2*c**4*x**2*sqrt(c + d*x**2)/(693*d**2)
+ b**2*c**3*x**4*sqrt(c + d*x**2)/(231*d) + 113*b**2*c**2*x**6*sqrt(c + d*x**2)/693 + 23*b**2*c*d*x**8*sqrt(c
+ d*x**2)/99 + b**2*d**2*x**10*sqrt(c + d*x**2)/11, Ne(d, 0)), (c**(5/2)*(a**2*x**2/2 + a*b*x**4/2 + b**2*x**6
/6), True))

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