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

Optimal. Leaf size=76 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (a d^2+2 b c^2\right )}{c^2 d^4}-\frac{x^2 \left (\frac{a}{c^2}+\frac{b}{d^2}\right )}{\sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

-(((a/c^2 + b/d^2)*x^2)/(Sqrt[-c + d*x]*Sqrt[c + d*x])) + ((2*b*c^2 + a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(c^
2*d^4)

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Rubi [A]  time = 0.0535191, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {458, 74} \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (a d^2+2 b c^2\right )}{c^2 d^4}-\frac{x^2 \left (\frac{a}{c^2}+\frac{b}{d^2}\right )}{\sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a/c^2 + b/d^2)*x^2)/(Sqrt[-c + d*x]*Sqrt[c + d*x])) + ((2*b*c^2 + a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(c^
2*d^4)

Rule 458

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> -Simp[((b1*b2*c - a1*a2*d)*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))
^(p + 1))/(a1*a2*b1*b2*e*n*(p + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*b1*b2*n*
(p + 1)), Int[(e*x)^m*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1), x], x] /; FreeQ[{a1, b1, a2, b2, c,
d, e, m, n}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -
5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1, m, -(n*(p + 1))]))

Rule 74

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

Rubi steps

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

Mathematica [A]  time = 0.0283267, size = 45, normalized size = 0.59 \[ \frac{-a d^2-2 b c^2+b d^2 x^2}{d^4 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*c^2 - a*d^2 + b*d^2*x^2)/(d^4*Sqrt[-c + d*x]*Sqrt[c + d*x])

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Maple [A]  time = 0.004, size = 43, normalized size = 0.6 \begin{align*} -{\frac{-b{d}^{2}{x}^{2}+a{d}^{2}+2\,b{c}^{2}}{{d}^{4}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-b*d^2*x^2+a*d^2+2*b*c^2)/(d*x+c)^(1/2)/d^4/(d*x-c)^(1/2)

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Maxima [A]  time = 0.966589, size = 93, normalized size = 1.22 \begin{align*} \frac{b x^{2}}{\sqrt{d^{2} x^{2} - c^{2}} d^{2}} - \frac{2 \, b c^{2}}{\sqrt{d^{2} x^{2} - c^{2}} d^{4}} - \frac{a}{\sqrt{d^{2} x^{2} - c^{2}} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x^2/(sqrt(d^2*x^2 - c^2)*d^2) - 2*b*c^2/(sqrt(d^2*x^2 - c^2)*d^4) - a/(sqrt(d^2*x^2 - c^2)*d^2)

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Fricas [A]  time = 1.50678, size = 107, normalized size = 1.41 \begin{align*} \frac{{\left (b d^{2} x^{2} - 2 \, b c^{2} - a d^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}}{d^{6} x^{2} - c^{2} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 52.8629, size = 201, normalized size = 2.64 \begin{align*} a \left (- \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4}, 1 & 0, 1, \frac{3}{2} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c d^{2}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, 1 & \\- \frac{1}{4}, \frac{1}{4} & -1, - \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c d^{2}}\right ) + b \left (\frac{c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & -1, 0, \frac{1}{2}, 1 \\- \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{4}} - \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & -2, - \frac{3}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{4}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-meijerg(((1/4, 3/4, 1), (0, 1, 3/2)), ((1/4, 1/2, 3/4, 1, 3/2), (0,)), c**2/(d**2*x**2))/(2*pi**(3/2)*c*d*
*2) - I*meijerg(((-1, -1/2, -1/4, 0, 1/4, 1), ()), ((-1/4, 1/4), (-1, -1/2, 1/2, 0)), c**2*exp_polar(2*I*pi)/(
d**2*x**2))/(2*pi**(3/2)*c*d**2)) + b*(c*meijerg(((-3/4, -1/4), (-1, 0, 1/2, 1)), ((-3/4, -1/2, -1/4, 0, 1/2,
0), ()), c**2/(d**2*x**2))/(2*pi**(3/2)*d**4) - I*c*meijerg(((-2, -3/2, -5/4, -1, -3/4, 1), ()), ((-5/4, -3/4)
, (-2, -3/2, -1/2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(2*pi**(3/2)*d**4))

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Giac [A]  time = 1.25029, size = 127, normalized size = 1.67 \begin{align*} \frac{{\left (2 \,{\left (d x + c\right )} b d^{8} - \frac{5 \, b c^{2} d^{8} + a d^{10}}{c}\right )} \sqrt{d x + c}}{32 \, \sqrt{d x - c}} + \frac{2 \,{\left (b c^{2} + a d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(2*(d*x + c)*b*d^8 - (5*b*c^2*d^8 + a*d^10)/c)*sqrt(d*x + c)/sqrt(d*x - c) + 2*(b*c^2 + a*d^2)/(((sqrt(d*
x + c) - sqrt(d*x - c))^2 + 2*c)*d^4)