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

Optimal. Leaf size=72 \[ \frac{2 b (b c-a d)}{d^3 \sqrt{c+d x^2}}-\frac{(b c-a d)^2}{3 d^3 \left (c+d x^2\right )^{3/2}}+\frac{b^2 \sqrt{c+d x^2}}{d^3} \]

[Out]

-(b*c - a*d)^2/(3*d^3*(c + d*x^2)^(3/2)) + (2*b*(b*c - a*d))/(d^3*Sqrt[c + d*x^2]) + (b^2*Sqrt[c + d*x^2])/d^3

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Rubi [A]  time = 0.0581318, antiderivative size = 72, normalized size of antiderivative = 1., 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} \[ \frac{2 b (b c-a d)}{d^3 \sqrt{c+d x^2}}-\frac{(b c-a d)^2}{3 d^3 \left (c+d x^2\right )^{3/2}}+\frac{b^2 \sqrt{c+d x^2}}{d^3} \]

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/(3*d^3*(c + d*x^2)^(3/2)) + (2*b*(b*c - a*d))/(d^3*Sqrt[c + d*x^2]) + (b^2*Sqrt[c + d*x^2])/d^3

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]

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])

Rubi steps

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

Mathematica [A]  time = 0.0379859, size = 67, normalized size = 0.93 \[ \frac{-a^2 d^2-2 a b d \left (2 c+3 d x^2\right )+b^2 \left (8 c^2+12 c d x^2+3 d^2 x^4\right )}{3 d^3 \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^2*d^2) - 2*a*b*d*(2*c + 3*d*x^2) + b^2*(8*c^2 + 12*c*d*x^2 + 3*d^2*x^4))/(3*d^3*(c + d*x^2)^(3/2))

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Maple [A]  time = 0.005, size = 68, normalized size = 0.9 \begin{align*} -{\frac{-3\,{b}^{2}{d}^{2}{x}^{4}+6\,ab{d}^{2}{x}^{2}-12\,{b}^{2}cd{x}^{2}+{a}^{2}{d}^{2}+4\,cabd-8\,{b}^{2}{c}^{2}}{3\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

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]

-1/3*(-3*b^2*d^2*x^4+6*a*b*d^2*x^2-12*b^2*c*d*x^2+a^2*d^2+4*a*b*c*d-8*b^2*c^2)/(d*x^2+c)^(3/2)/d^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.3561, size = 182, normalized size = 2.53 \begin{align*} \frac{{\left (3 \, b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 4 \, a b c d - a^{2} d^{2} + 6 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} \end{align*}

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]

1/3*(3*b^2*d^2*x^4 + 8*b^2*c^2 - 4*a*b*c*d - a^2*d^2 + 6*(2*b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)/(d^5*x^4 +
2*c*d^4*x^2 + c^2*d^3)

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Sympy [A]  time = 1.27729, size = 303, normalized size = 4.21 \begin{align*} \begin{cases} - \frac{a^{2} d^{2}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} - \frac{4 a b c d}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} - \frac{6 a b d^{2} x^{2}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} + \frac{8 b^{2} c^{2}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} + \frac{12 b^{2} c d x^{2}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} + \frac{3 b^{2} d^{2} x^{4}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}}{c^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

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*d**2/(3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2)) - 4*a*b*c*d/(3*c*d**3*sqrt(c
+ d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2)) - 6*a*b*d**2*x**2/(3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c +
d*x**2)) + 8*b**2*c**2/(3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2)) + 12*b**2*c*d*x**2/(3*c*d**3
*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2)) + 3*b**2*d**2*x**4/(3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*
sqrt(c + d*x**2)), Ne(d, 0)), ((a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6)/c**(5/2), True))

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Giac [A]  time = 1.16322, size = 105, normalized size = 1.46 \begin{align*} \frac{3 \, \sqrt{d x^{2} + c} b^{2} + \frac{6 \,{\left (d x^{2} + c\right )} b^{2} c - b^{2} c^{2} - 6 \,{\left (d x^{2} + c\right )} a b d + 2 \, a b c d - a^{2} d^{2}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}}{3 \, d^{3}} \end{align*}

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]

1/3*(3*sqrt(d*x^2 + c)*b^2 + (6*(d*x^2 + c)*b^2*c - b^2*c^2 - 6*(d*x^2 + c)*a*b*d + 2*a*b*c*d - a^2*d^2)/(d*x^
2 + c)^(3/2))/d^3