3.616 \(\int x (a+b x^2)^2 (c+d x^2)^{3/2} \, dx\)

Optimal. Leaf size=77 \[ -\frac {2 b \left (c+d x^2\right )^{7/2} (b c-a d)}{7 d^3}+\frac {\left (c+d x^2\right )^{5/2} (b c-a d)^2}{5 d^3}+\frac {b^2 \left (c+d x^2\right )^{9/2}}{9 d^3} \]

[Out]

1/5*(-a*d+b*c)^2*(d*x^2+c)^(5/2)/d^3-2/7*b*(-a*d+b*c)*(d*x^2+c)^(7/2)/d^3+1/9*b^2*(d*x^2+c)^(9/2)/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} \[ -\frac {2 b \left (c+d x^2\right )^{7/2} (b c-a d)}{7 d^3}+\frac {\left (c+d x^2\right )^{5/2} (b c-a d)^2}{5 d^3}+\frac {b^2 \left (c+d x^2\right )^{9/2}}{9 d^3} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]

[Out]

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

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Mathematica [A]  time = 0.05, size = 67, normalized size = 0.87 \[ \frac {\left (c+d x^2\right )^{5/2} \left (63 a^2 d^2+18 a b d \left (5 d x^2-2 c\right )+b^2 \left (8 c^2-20 c d x^2+35 d^2 x^4\right )\right )}{315 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]

[Out]

((c + d*x^2)^(5/2)*(63*a^2*d^2 + 18*a*b*d*(-2*c + 5*d*x^2) + b^2*(8*c^2 - 20*c*d*x^2 + 35*d^2*x^4)))/(315*d^3)

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fricas [B]  time = 0.79, size = 141, normalized size = 1.83 \[ \frac {{\left (35 \, b^{2} d^{4} x^{8} + 10 \, {\left (5 \, b^{2} c d^{3} + 9 \, a b d^{4}\right )} x^{6} + 8 \, b^{2} c^{4} - 36 \, a b c^{3} d + 63 \, a^{2} c^{2} d^{2} + 3 \, {\left (b^{2} c^{2} d^{2} + 48 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{4} - 2 \, {\left (2 \, b^{2} c^{3} d - 9 \, a b c^{2} d^{2} - 63 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{315 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^2*(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

1/315*(35*b^2*d^4*x^8 + 10*(5*b^2*c*d^3 + 9*a*b*d^4)*x^6 + 8*b^2*c^4 - 36*a*b*c^3*d + 63*a^2*c^2*d^2 + 3*(b^2*
c^2*d^2 + 48*a*b*c*d^3 + 21*a^2*d^4)*x^4 - 2*(2*b^2*c^3*d - 9*a*b*c^2*d^2 - 63*a^2*c*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 \[ \frac {35 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} - 90 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c + 63 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} + 90 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d - 126 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c d + 63 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{2}}{315 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^2*(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

1/315*(35*(d*x^2 + c)^(9/2)*b^2 - 90*(d*x^2 + c)^(7/2)*b^2*c + 63*(d*x^2 + c)^(5/2)*b^2*c^2 + 90*(d*x^2 + c)^(
7/2)*a*b*d - 126*(d*x^2 + c)^(5/2)*a*b*c*d + 63*(d*x^2 + c)^(5/2)*a^2*d^2)/d^3

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maple [A]  time = 0.01, size = 69, normalized size = 0.90 \[ \frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} \left (35 b^{2} x^{4} d^{2}+90 a b \,d^{2} x^{2}-20 b^{2} c d \,x^{2}+63 a^{2} d^{2}-36 a b c d +8 b^{2} c^{2}\right )}{315 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^2*(d*x^2+c)^(3/2),x)

[Out]

1/315*(d*x^2+c)^(5/2)*(35*b^2*d^2*x^4+90*a*b*d^2*x^2-20*b^2*c*d*x^2+63*a^2*d^2-36*a*b*c*d+8*b^2*c^2)/d^3

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maxima [A]  time = 1.00, size = 115, normalized size = 1.49 \[ \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x^{4}}{9 \, d} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c x^{2}}{63 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b x^{2}}{7 \, d} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2}}{315 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c}{35 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2}}{5 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^2*(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

1/9*(d*x^2 + c)^(5/2)*b^2*x^4/d - 4/63*(d*x^2 + c)^(5/2)*b^2*c*x^2/d^2 + 2/7*(d*x^2 + c)^(5/2)*a*b*x^2/d + 8/3
15*(d*x^2 + c)^(5/2)*b^2*c^2/d^3 - 4/35*(d*x^2 + c)^(5/2)*a*b*c/d^2 + 1/5*(d*x^2 + c)^(5/2)*a^2/d

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mupad [B]  time = 0.73, size = 136, normalized size = 1.77 \[ \sqrt {d\,x^2+c}\,\left (\frac {63\,a^2\,c^2\,d^2-36\,a\,b\,c^3\,d+8\,b^2\,c^4}{315\,d^3}+\frac {x^4\,\left (63\,a^2\,d^4+144\,a\,b\,c\,d^3+3\,b^2\,c^2\,d^2\right )}{315\,d^3}+\frac {2\,b\,x^6\,\left (9\,a\,d+5\,b\,c\right )}{63}+\frac {b^2\,d\,x^8}{9}+\frac {2\,c\,x^2\,\left (63\,a^2\,d^2+9\,a\,b\,c\,d-2\,b^2\,c^2\right )}{315\,d^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^2*(c + d*x^2)^(3/2),x)

[Out]

(c + d*x^2)^(1/2)*((8*b^2*c^4 + 63*a^2*c^2*d^2 - 36*a*b*c^3*d)/(315*d^3) + (x^4*(63*a^2*d^4 + 3*b^2*c^2*d^2 +
144*a*b*c*d^3))/(315*d^3) + (2*b*x^6*(9*a*d + 5*b*c))/63 + (b^2*d*x^8)/9 + (2*c*x^2*(63*a^2*d^2 - 2*b^2*c^2 +
9*a*b*c*d))/(315*d^2))

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sympy [A]  time = 3.68, size = 303, normalized size = 3.94 \[ \begin {cases} \frac {a^{2} c^{2} \sqrt {c + d x^{2}}}{5 d} + \frac {2 a^{2} c x^{2} \sqrt {c + d x^{2}}}{5} + \frac {a^{2} d x^{4} \sqrt {c + d x^{2}}}{5} - \frac {4 a b c^{3} \sqrt {c + d x^{2}}}{35 d^{2}} + \frac {2 a b c^{2} x^{2} \sqrt {c + d x^{2}}}{35 d} + \frac {16 a b c x^{4} \sqrt {c + d x^{2}}}{35} + \frac {2 a b d x^{6} \sqrt {c + d x^{2}}}{7} + \frac {8 b^{2} c^{4} \sqrt {c + d x^{2}}}{315 d^{3}} - \frac {4 b^{2} c^{3} x^{2} \sqrt {c + d x^{2}}}{315 d^{2}} + \frac {b^{2} c^{2} x^{4} \sqrt {c + d x^{2}}}{105 d} + \frac {10 b^{2} c x^{6} \sqrt {c + d x^{2}}}{63} + \frac {b^{2} d x^{8} \sqrt {c + d x^{2}}}{9} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a)**2*(d*x**2+c)**(3/2),x)

[Out]

Piecewise((a**2*c**2*sqrt(c + d*x**2)/(5*d) + 2*a**2*c*x**2*sqrt(c + d*x**2)/5 + a**2*d*x**4*sqrt(c + d*x**2)/
5 - 4*a*b*c**3*sqrt(c + d*x**2)/(35*d**2) + 2*a*b*c**2*x**2*sqrt(c + d*x**2)/(35*d) + 16*a*b*c*x**4*sqrt(c + d
*x**2)/35 + 2*a*b*d*x**6*sqrt(c + d*x**2)/7 + 8*b**2*c**4*sqrt(c + d*x**2)/(315*d**3) - 4*b**2*c**3*x**2*sqrt(
c + d*x**2)/(315*d**2) + b**2*c**2*x**4*sqrt(c + d*x**2)/(105*d) + 10*b**2*c*x**6*sqrt(c + d*x**2)/63 + b**2*d
*x**8*sqrt(c + d*x**2)/9, Ne(d, 0)), (c**(3/2)*(a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6), True))

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