3.97 \(\int \frac{(a+b x^2)^3}{(c+d x^2)^{11/2}} \, dx\)

Optimal. Leaf size=224 \[ \frac{8 a^2 x \left (a+b x^2\right ) (9 b c-8 a d)}{315 c^4 \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt{c+d x^2} (b c-a d)}+\frac{x \left (a+b x^2\right )^3 (9 b c-8 a d)}{63 c^2 \left (c+d x^2\right )^{7/2} (b c-a d)}+\frac{2 a x \left (a+b x^2\right )^2 (9 b c-8 a d)}{105 c^3 \left (c+d x^2\right )^{5/2} (b c-a d)}-\frac{d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)} \]

[Out]

-(d*x*(a + b*x^2)^4)/(9*c*(b*c - a*d)*(c + d*x^2)^(9/2)) + ((9*b*c - 8*a*d)*x*(a + b*x^2)^3)/(63*c^2*(b*c - a*
d)*(c + d*x^2)^(7/2)) + (2*a*(9*b*c - 8*a*d)*x*(a + b*x^2)^2)/(105*c^3*(b*c - a*d)*(c + d*x^2)^(5/2)) + (8*a^2
*(9*b*c - 8*a*d)*x*(a + b*x^2))/(315*c^4*(b*c - a*d)*(c + d*x^2)^(3/2)) + (16*a^3*(9*b*c - 8*a*d)*x)/(315*c^5*
(b*c - a*d)*Sqrt[c + d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.101163, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {382, 378, 191} \[ \frac{8 a^2 x \left (a+b x^2\right ) (9 b c-8 a d)}{315 c^4 \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt{c+d x^2} (b c-a d)}+\frac{x \left (a+b x^2\right )^3 (9 b c-8 a d)}{63 c^2 \left (c+d x^2\right )^{7/2} (b c-a d)}+\frac{2 a x \left (a+b x^2\right )^2 (9 b c-8 a d)}{105 c^3 \left (c+d x^2\right )^{5/2} (b c-a d)}-\frac{d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/(c + d*x^2)^(11/2),x]

[Out]

-(d*x*(a + b*x^2)^4)/(9*c*(b*c - a*d)*(c + d*x^2)^(9/2)) + ((9*b*c - 8*a*d)*x*(a + b*x^2)^3)/(63*c^2*(b*c - a*
d)*(c + d*x^2)^(7/2)) + (2*a*(9*b*c - 8*a*d)*x*(a + b*x^2)^2)/(105*c^3*(b*c - a*d)*(c + d*x^2)^(5/2)) + (8*a^2
*(9*b*c - 8*a*d)*x*(a + b*x^2))/(315*c^4*(b*c - a*d)*(c + d*x^2)^(3/2)) + (16*a^3*(9*b*c - 8*a*d)*x)/(315*c^5*
(b*c - a*d)*Sqrt[c + d*x^2])

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 378

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx &=-\frac{d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac{(9 b c-8 a d) \int \frac{\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{9/2}} \, dx}{9 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac{(9 b c-8 a d) x \left (a+b x^2\right )^3}{63 c^2 (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac{(2 a (9 b c-8 a d)) \int \frac{\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx}{21 c^2 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac{(9 b c-8 a d) x \left (a+b x^2\right )^3}{63 c^2 (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac{2 a (9 b c-8 a d) x \left (a+b x^2\right )^2}{105 c^3 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac{\left (8 a^2 (9 b c-8 a d)\right ) \int \frac{a+b x^2}{\left (c+d x^2\right )^{5/2}} \, dx}{105 c^3 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac{(9 b c-8 a d) x \left (a+b x^2\right )^3}{63 c^2 (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac{2 a (9 b c-8 a d) x \left (a+b x^2\right )^2}{105 c^3 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac{8 a^2 (9 b c-8 a d) x \left (a+b x^2\right )}{315 c^4 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{\left (16 a^3 (9 b c-8 a d)\right ) \int \frac{1}{\left (c+d x^2\right )^{3/2}} \, dx}{315 c^4 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac{(9 b c-8 a d) x \left (a+b x^2\right )^3}{63 c^2 (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac{2 a (9 b c-8 a d) x \left (a+b x^2\right )^2}{105 c^3 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac{8 a^2 (9 b c-8 a d) x \left (a+b x^2\right )}{315 c^4 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{16 a^3 (9 b c-8 a d) x}{315 c^5 (b c-a d) \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.10089, size = 163, normalized size = 0.73 \[ \frac{3 a^2 b c x^3 \left (126 c^2 d x^2+105 c^3+72 c d^2 x^4+16 d^3 x^6\right )+a^3 \left (1008 c^2 d^2 x^5+840 c^3 d x^3+315 c^4 x+576 c d^3 x^7+128 d^4 x^9\right )+3 a b^2 c^2 x^5 \left (63 c^2+36 c d x^2+8 d^2 x^4\right )+5 b^3 c^3 x^7 \left (9 c+2 d x^2\right )}{315 c^5 \left (c+d x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/(c + d*x^2)^(11/2),x]

[Out]

(5*b^3*c^3*x^7*(9*c + 2*d*x^2) + 3*a*b^2*c^2*x^5*(63*c^2 + 36*c*d*x^2 + 8*d^2*x^4) + 3*a^2*b*c*x^3*(105*c^3 +
126*c^2*d*x^2 + 72*c*d^2*x^4 + 16*d^3*x^6) + a^3*(315*c^4*x + 840*c^3*d*x^3 + 1008*c^2*d^2*x^5 + 576*c*d^3*x^7
+ 128*d^4*x^9))/(315*c^5*(c + d*x^2)^(9/2))

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Maple [A]  time = 0.007, size = 190, normalized size = 0.9 \begin{align*}{\frac{x \left ( 128\,{a}^{3}{d}^{4}{x}^{8}+48\,{a}^{2}bc{d}^{3}{x}^{8}+24\,a{b}^{2}{c}^{2}{d}^{2}{x}^{8}+10\,{b}^{3}{c}^{3}d{x}^{8}+576\,{a}^{3}c{d}^{3}{x}^{6}+216\,{a}^{2}b{c}^{2}{d}^{2}{x}^{6}+108\,a{b}^{2}{c}^{3}d{x}^{6}+45\,{b}^{3}{c}^{4}{x}^{6}+1008\,{a}^{3}{c}^{2}{d}^{2}{x}^{4}+378\,{a}^{2}b{c}^{3}d{x}^{4}+189\,a{b}^{2}{c}^{4}{x}^{4}+840\,{a}^{3}{c}^{3}d{x}^{2}+315\,{a}^{2}b{c}^{4}{x}^{2}+315\,{a}^{3}{c}^{4} \right ) }{315\,{c}^{5}} \left ( d{x}^{2}+c \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/(d*x^2+c)^(11/2),x)

[Out]

1/315*x*(128*a^3*d^4*x^8+48*a^2*b*c*d^3*x^8+24*a*b^2*c^2*d^2*x^8+10*b^3*c^3*d*x^8+576*a^3*c*d^3*x^6+216*a^2*b*
c^2*d^2*x^6+108*a*b^2*c^3*d*x^6+45*b^3*c^4*x^6+1008*a^3*c^2*d^2*x^4+378*a^2*b*c^3*d*x^4+189*a*b^2*c^4*x^4+840*
a^3*c^3*d*x^2+315*a^2*b*c^4*x^2+315*a^3*c^4)/(d*x^2+c)^(9/2)/c^5

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Maxima [B]  time = 1.05908, size = 628, normalized size = 2.8 \begin{align*} -\frac{b^{3} x^{5}}{4 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d} - \frac{5 \, b^{3} c x^{3}}{24 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d^{2}} - \frac{a b^{2} x^{3}}{2 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d} + \frac{128 \, a^{3} x}{315 \, \sqrt{d x^{2} + c} c^{5}} + \frac{64 \, a^{3} x}{315 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{4}} + \frac{16 \, a^{3} x}{105 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c^{3}} + \frac{8 \, a^{3} x}{63 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} c^{2}} + \frac{a^{3} x}{9 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} c} + \frac{b^{3} x}{84 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} d^{3}} + \frac{2 \, b^{3} x}{63 \, \sqrt{d x^{2} + c} c^{2} d^{3}} + \frac{b^{3} x}{63 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c d^{3}} + \frac{5 \, b^{3} c x}{504 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} d^{3}} - \frac{5 \, b^{3} c^{2} x}{72 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d^{3}} + \frac{a b^{2} x}{42 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} d^{2}} + \frac{8 \, a b^{2} x}{105 \, \sqrt{d x^{2} + c} c^{3} d^{2}} + \frac{4 \, a b^{2} x}{105 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{2} d^{2}} + \frac{a b^{2} x}{35 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c d^{2}} - \frac{a b^{2} c x}{6 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d^{2}} - \frac{a^{2} b x}{3 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d} + \frac{16 \, a^{2} b x}{105 \, \sqrt{d x^{2} + c} c^{4} d} + \frac{8 \, a^{2} b x}{105 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{3} d} + \frac{2 \, a^{2} b x}{35 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c^{2} d} + \frac{a^{2} b x}{21 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/(d*x^2+c)^(11/2),x, algorithm="maxima")

[Out]

-1/4*b^3*x^5/((d*x^2 + c)^(9/2)*d) - 5/24*b^3*c*x^3/((d*x^2 + c)^(9/2)*d^2) - 1/2*a*b^2*x^3/((d*x^2 + c)^(9/2)
*d) + 128/315*a^3*x/(sqrt(d*x^2 + c)*c^5) + 64/315*a^3*x/((d*x^2 + c)^(3/2)*c^4) + 16/105*a^3*x/((d*x^2 + c)^(
5/2)*c^3) + 8/63*a^3*x/((d*x^2 + c)^(7/2)*c^2) + 1/9*a^3*x/((d*x^2 + c)^(9/2)*c) + 1/84*b^3*x/((d*x^2 + c)^(5/
2)*d^3) + 2/63*b^3*x/(sqrt(d*x^2 + c)*c^2*d^3) + 1/63*b^3*x/((d*x^2 + c)^(3/2)*c*d^3) + 5/504*b^3*c*x/((d*x^2
+ c)^(7/2)*d^3) - 5/72*b^3*c^2*x/((d*x^2 + c)^(9/2)*d^3) + 1/42*a*b^2*x/((d*x^2 + c)^(7/2)*d^2) + 8/105*a*b^2*
x/(sqrt(d*x^2 + c)*c^3*d^2) + 4/105*a*b^2*x/((d*x^2 + c)^(3/2)*c^2*d^2) + 1/35*a*b^2*x/((d*x^2 + c)^(5/2)*c*d^
2) - 1/6*a*b^2*c*x/((d*x^2 + c)^(9/2)*d^2) - 1/3*a^2*b*x/((d*x^2 + c)^(9/2)*d) + 16/105*a^2*b*x/(sqrt(d*x^2 +
c)*c^4*d) + 8/105*a^2*b*x/((d*x^2 + c)^(3/2)*c^3*d) + 2/35*a^2*b*x/((d*x^2 + c)^(5/2)*c^2*d) + 1/21*a^2*b*x/((
d*x^2 + c)^(7/2)*c*d)

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Fricas [A]  time = 3.34891, size = 487, normalized size = 2.17 \begin{align*} \frac{{\left (2 \,{\left (5 \, b^{3} c^{3} d + 12 \, a b^{2} c^{2} d^{2} + 24 \, a^{2} b c d^{3} + 64 \, a^{3} d^{4}\right )} x^{9} + 315 \, a^{3} c^{4} x + 9 \,{\left (5 \, b^{3} c^{4} + 12 \, a b^{2} c^{3} d + 24 \, a^{2} b c^{2} d^{2} + 64 \, a^{3} c d^{3}\right )} x^{7} + 63 \,{\left (3 \, a b^{2} c^{4} + 6 \, a^{2} b c^{3} d + 16 \, a^{3} c^{2} d^{2}\right )} x^{5} + 105 \,{\left (3 \, a^{2} b c^{4} + 8 \, a^{3} c^{3} d\right )} x^{3}\right )} \sqrt{d x^{2} + c}}{315 \,{\left (c^{5} d^{5} x^{10} + 5 \, c^{6} d^{4} x^{8} + 10 \, c^{7} d^{3} x^{6} + 10 \, c^{8} d^{2} x^{4} + 5 \, c^{9} d x^{2} + c^{10}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/(d*x^2+c)^(11/2),x, algorithm="fricas")

[Out]

1/315*(2*(5*b^3*c^3*d + 12*a*b^2*c^2*d^2 + 24*a^2*b*c*d^3 + 64*a^3*d^4)*x^9 + 315*a^3*c^4*x + 9*(5*b^3*c^4 + 1
2*a*b^2*c^3*d + 24*a^2*b*c^2*d^2 + 64*a^3*c*d^3)*x^7 + 63*(3*a*b^2*c^4 + 6*a^2*b*c^3*d + 16*a^3*c^2*d^2)*x^5 +
105*(3*a^2*b*c^4 + 8*a^3*c^3*d)*x^3)*sqrt(d*x^2 + c)/(c^5*d^5*x^10 + 5*c^6*d^4*x^8 + 10*c^7*d^3*x^6 + 10*c^8*
d^2*x^4 + 5*c^9*d*x^2 + c^10)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/(d*x**2+c)**(11/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.17216, size = 294, normalized size = 1.31 \begin{align*} \frac{{\left ({\left ({\left (x^{2}{\left (\frac{2 \,{\left (5 \, b^{3} c^{3} d^{5} + 12 \, a b^{2} c^{2} d^{6} + 24 \, a^{2} b c d^{7} + 64 \, a^{3} d^{8}\right )} x^{2}}{c^{5} d^{4}} + \frac{9 \,{\left (5 \, b^{3} c^{4} d^{4} + 12 \, a b^{2} c^{3} d^{5} + 24 \, a^{2} b c^{2} d^{6} + 64 \, a^{3} c d^{7}\right )}}{c^{5} d^{4}}\right )} + \frac{63 \,{\left (3 \, a b^{2} c^{4} d^{4} + 6 \, a^{2} b c^{3} d^{5} + 16 \, a^{3} c^{2} d^{6}\right )}}{c^{5} d^{4}}\right )} x^{2} + \frac{105 \,{\left (3 \, a^{2} b c^{4} d^{4} + 8 \, a^{3} c^{3} d^{5}\right )}}{c^{5} d^{4}}\right )} x^{2} + \frac{315 \, a^{3}}{c}\right )} x}{315 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/(d*x^2+c)^(11/2),x, algorithm="giac")

[Out]

1/315*(((x^2*(2*(5*b^3*c^3*d^5 + 12*a*b^2*c^2*d^6 + 24*a^2*b*c*d^7 + 64*a^3*d^8)*x^2/(c^5*d^4) + 9*(5*b^3*c^4*
d^4 + 12*a*b^2*c^3*d^5 + 24*a^2*b*c^2*d^6 + 64*a^3*c*d^7)/(c^5*d^4)) + 63*(3*a*b^2*c^4*d^4 + 6*a^2*b*c^3*d^5 +
16*a^3*c^2*d^6)/(c^5*d^4))*x^2 + 105*(3*a^2*b*c^4*d^4 + 8*a^3*c^3*d^5)/(c^5*d^4))*x^2 + 315*a^3/c)*x/(d*x^2 +
c)^(9/2)