3.98 \(\int \frac{(a+b x^2)^2}{(c+d x^2)^{9/2}} \, dx\)

Optimal. Leaf size=174 \[ \frac{8 a^2 x (7 b c-6 a d)}{105 c^4 \sqrt{c+d x^2} (b c-a d)}+\frac{x \left (a+b x^2\right )^2 (7 b c-6 a d)}{35 c^2 \left (c+d x^2\right )^{5/2} (b c-a d)}+\frac{4 a x \left (a+b x^2\right ) (7 b c-6 a d)}{105 c^3 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{d x \left (a+b x^2\right )^3}{7 c \left (c+d x^2\right )^{7/2} (b c-a d)} \]

[Out]

-(d*x*(a + b*x^2)^3)/(7*c*(b*c - a*d)*(c + d*x^2)^(7/2)) + ((7*b*c - 6*a*d)*x*(a + b*x^2)^2)/(35*c^2*(b*c - a*
d)*(c + d*x^2)^(5/2)) + (4*a*(7*b*c - 6*a*d)*x*(a + b*x^2))/(105*c^3*(b*c - a*d)*(c + d*x^2)^(3/2)) + (8*a^2*(
7*b*c - 6*a*d)*x)/(105*c^4*(b*c - a*d)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.0707433, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {382, 378, 191} \[ \frac{8 a^2 x (7 b c-6 a d)}{105 c^4 \sqrt{c+d x^2} (b c-a d)}+\frac{x \left (a+b x^2\right )^2 (7 b c-6 a d)}{35 c^2 \left (c+d x^2\right )^{5/2} (b c-a d)}+\frac{4 a x \left (a+b x^2\right ) (7 b c-6 a d)}{105 c^3 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{d x \left (a+b x^2\right )^3}{7 c \left (c+d x^2\right )^{7/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(c + d*x^2)^(9/2),x]

[Out]

-(d*x*(a + b*x^2)^3)/(7*c*(b*c - a*d)*(c + d*x^2)^(7/2)) + ((7*b*c - 6*a*d)*x*(a + b*x^2)^2)/(35*c^2*(b*c - a*
d)*(c + d*x^2)^(5/2)) + (4*a*(7*b*c - 6*a*d)*x*(a + b*x^2))/(105*c^3*(b*c - a*d)*(c + d*x^2)^(3/2)) + (8*a^2*(
7*b*c - 6*a*d)*x)/(105*c^4*(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 )^2}{\left (c+d x^2\right )^{9/2}} \, dx &=-\frac{d x \left (a+b x^2\right )^3}{7 c (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac{(7 b c-6 a d) \int \frac{\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx}{7 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^3}{7 c (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac{(7 b c-6 a d) x \left (a+b x^2\right )^2}{35 c^2 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac{(4 a (7 b c-6 a d)) \int \frac{a+b x^2}{\left (c+d x^2\right )^{5/2}} \, dx}{35 c^2 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^3}{7 c (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac{(7 b c-6 a d) x \left (a+b x^2\right )^2}{35 c^2 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac{4 a (7 b c-6 a d) x \left (a+b x^2\right )}{105 c^3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{\left (8 a^2 (7 b c-6 a d)\right ) \int \frac{1}{\left (c+d x^2\right )^{3/2}} \, dx}{105 c^3 (b c-a d)}\\ &=-\frac{d x \left (a+b x^2\right )^3}{7 c (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac{(7 b c-6 a d) x \left (a+b x^2\right )^2}{35 c^2 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac{4 a (7 b c-6 a d) x \left (a+b x^2\right )}{105 c^3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{8 a^2 (7 b c-6 a d) x}{105 c^4 (b c-a d) \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.063847, size = 107, normalized size = 0.61 \[ \frac{3 a^2 \left (70 c^2 d x^3+35 c^3 x+56 c d^2 x^5+16 d^3 x^7\right )+2 a b c x^3 \left (35 c^2+28 c d x^2+8 d^2 x^4\right )+3 b^2 c^2 x^5 \left (7 c+2 d x^2\right )}{105 c^4 \left (c+d x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/(c + d*x^2)^(9/2),x]

[Out]

(3*b^2*c^2*x^5*(7*c + 2*d*x^2) + 2*a*b*c*x^3*(35*c^2 + 28*c*d*x^2 + 8*d^2*x^4) + 3*a^2*(35*c^3*x + 70*c^2*d*x^
3 + 56*c*d^2*x^5 + 16*d^3*x^7))/(105*c^4*(c + d*x^2)^(7/2))

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Maple [A]  time = 0.006, size = 115, normalized size = 0.7 \begin{align*}{\frac{x \left ( 48\,{a}^{2}{d}^{3}{x}^{6}+16\,abc{d}^{2}{x}^{6}+6\,{b}^{2}{c}^{2}d{x}^{6}+168\,{a}^{2}c{d}^{2}{x}^{4}+56\,ab{c}^{2}d{x}^{4}+21\,{b}^{2}{c}^{3}{x}^{4}+210\,{a}^{2}{c}^{2}d{x}^{2}+70\,ab{c}^{3}{x}^{2}+105\,{a}^{2}{c}^{3} \right ) }{105\,{c}^{4}} \left ( d{x}^{2}+c \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/(d*x^2+c)^(9/2),x)

[Out]

1/105*x*(48*a^2*d^3*x^6+16*a*b*c*d^2*x^6+6*b^2*c^2*d*x^6+168*a^2*c*d^2*x^4+56*a*b*c^2*d*x^4+21*b^2*c^3*x^4+210
*a^2*c^2*d*x^2+70*a*b*c^3*x^2+105*a^2*c^3)/(d*x^2+c)^(7/2)/c^4

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Maxima [A]  time = 1.01433, size = 336, normalized size = 1.93 \begin{align*} -\frac{b^{2} x^{3}}{4 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} d} + \frac{16 \, a^{2} x}{35 \, \sqrt{d x^{2} + c} c^{4}} + \frac{8 \, a^{2} x}{35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{3}} + \frac{6 \, a^{2} x}{35 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c^{2}} + \frac{a^{2} x}{7 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} c} + \frac{3 \, b^{2} x}{140 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} d^{2}} + \frac{2 \, b^{2} x}{35 \, \sqrt{d x^{2} + c} c^{2} d^{2}} + \frac{b^{2} x}{35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c d^{2}} - \frac{3 \, b^{2} c x}{28 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} d^{2}} - \frac{2 \, a b x}{7 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} d} + \frac{16 \, a b x}{105 \, \sqrt{d x^{2} + c} c^{3} d} + \frac{8 \, a b x}{105 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{2} d} + \frac{2 \, a b x}{35 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^(9/2),x, algorithm="maxima")

[Out]

-1/4*b^2*x^3/((d*x^2 + c)^(7/2)*d) + 16/35*a^2*x/(sqrt(d*x^2 + c)*c^4) + 8/35*a^2*x/((d*x^2 + c)^(3/2)*c^3) +
6/35*a^2*x/((d*x^2 + c)^(5/2)*c^2) + 1/7*a^2*x/((d*x^2 + c)^(7/2)*c) + 3/140*b^2*x/((d*x^2 + c)^(5/2)*d^2) + 2
/35*b^2*x/(sqrt(d*x^2 + c)*c^2*d^2) + 1/35*b^2*x/((d*x^2 + c)^(3/2)*c*d^2) - 3/28*b^2*c*x/((d*x^2 + c)^(7/2)*d
^2) - 2/7*a*b*x/((d*x^2 + c)^(7/2)*d) + 16/105*a*b*x/(sqrt(d*x^2 + c)*c^3*d) + 8/105*a*b*x/((d*x^2 + c)^(3/2)*
c^2*d) + 2/35*a*b*x/((d*x^2 + c)^(5/2)*c*d)

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Fricas [A]  time = 1.94563, size = 319, normalized size = 1.83 \begin{align*} \frac{{\left (2 \,{\left (3 \, b^{2} c^{2} d + 8 \, a b c d^{2} + 24 \, a^{2} d^{3}\right )} x^{7} + 105 \, a^{2} c^{3} x + 7 \,{\left (3 \, b^{2} c^{3} + 8 \, a b c^{2} d + 24 \, a^{2} c d^{2}\right )} x^{5} + 70 \,{\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3}\right )} \sqrt{d x^{2} + c}}{105 \,{\left (c^{4} d^{4} x^{8} + 4 \, c^{5} d^{3} x^{6} + 6 \, c^{6} d^{2} x^{4} + 4 \, c^{7} d x^{2} + c^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^(9/2),x, algorithm="fricas")

[Out]

1/105*(2*(3*b^2*c^2*d + 8*a*b*c*d^2 + 24*a^2*d^3)*x^7 + 105*a^2*c^3*x + 7*(3*b^2*c^3 + 8*a*b*c^2*d + 24*a^2*c*
d^2)*x^5 + 70*(a*b*c^3 + 3*a^2*c^2*d)*x^3)*sqrt(d*x^2 + c)/(c^4*d^4*x^8 + 4*c^5*d^3*x^6 + 6*c^6*d^2*x^4 + 4*c^
7*d*x^2 + c^8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac{9}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/(d*x**2+c)**(9/2),x)

[Out]

Integral((a + b*x**2)**2/(c + d*x**2)**(9/2), x)

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Giac [A]  time = 1.25454, size = 186, normalized size = 1.07 \begin{align*} \frac{{\left ({\left (x^{2}{\left (\frac{2 \,{\left (3 \, b^{2} c^{2} d^{4} + 8 \, a b c d^{5} + 24 \, a^{2} d^{6}\right )} x^{2}}{c^{4} d^{3}} + \frac{7 \,{\left (3 \, b^{2} c^{3} d^{3} + 8 \, a b c^{2} d^{4} + 24 \, a^{2} c d^{5}\right )}}{c^{4} d^{3}}\right )} + \frac{70 \,{\left (a b c^{3} d^{3} + 3 \, a^{2} c^{2} d^{4}\right )}}{c^{4} d^{3}}\right )} x^{2} + \frac{105 \, a^{2}}{c}\right )} x}{105 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^(9/2),x, algorithm="giac")

[Out]

1/105*((x^2*(2*(3*b^2*c^2*d^4 + 8*a*b*c*d^5 + 24*a^2*d^6)*x^2/(c^4*d^3) + 7*(3*b^2*c^3*d^3 + 8*a*b*c^2*d^4 + 2
4*a^2*c*d^5)/(c^4*d^3)) + 70*(a*b*c^3*d^3 + 3*a^2*c^2*d^4)/(c^4*d^3))*x^2 + 105*a^2/c)*x/(d*x^2 + c)^(7/2)