3.583 \(\int \frac{(d+e x)^3 (f+g x)}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{(d+e x)^3 (d g+e f)}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+e x) (2 e f-3 d g)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (2 e f-3 d g)}{15 d^3 e \sqrt{d^2-e^2 x^2}} \]

[Out]

((e*f + d*g)*(d + e*x)^3)/(5*d*e^2*(d^2 - e^2*x^2)^(5/2)) + (2*(2*e*f - 3*d*g)*(d + e*x))/(15*d*e^2*(d^2 - e^2
*x^2)^(3/2)) + ((2*e*f - 3*d*g)*x)/(15*d^3*e*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0593807, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {789, 653, 191} \[ \frac{(d+e x)^3 (d g+e f)}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+e x) (2 e f-3 d g)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (2 e f-3 d g)}{15 d^3 e \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(f + g*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((e*f + d*g)*(d + e*x)^3)/(5*d*e^2*(d^2 - e^2*x^2)^(5/2)) + (2*(2*e*f - 3*d*g)*(d + e*x))/(15*d*e^2*(d^2 - e^2
*x^2)^(3/2)) + ((2*e*f - 3*d*g)*x)/(15*d^3*e*Sqrt[d^2 - e^2*x^2])

Rule 789

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d*g + e*f)*
(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(p + 1)), x] - Dist[(e*(m*(d*g + e*f) + 2*e*f*(p + 1)))/(2*c*d*(p + 1)
), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0]
&& LtQ[p, -1] && GtQ[m, 0]

Rule 653

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + c*x^2)^(p + 1))/(c*(
p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[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{(d+e x)^3 (f+g x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{(-5 e f+3 (e f+d g)) \int \frac{(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d e}\\ &=\frac{(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (2 e f-3 d g) (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{(-5 e f+3 (e f+d g)) \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e}\\ &=\frac{(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (2 e f-3 d g) (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{(2 e f-3 d g) x}{15 d^3 e \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.234674, size = 83, normalized size = 0.71 \[ -\frac{(d+e x) \left (-d^2 e (7 f+9 g x)+3 d^3 g+3 d e^2 x (2 f+g x)-2 e^3 f x^2\right )}{15 d^3 e^2 (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(f + g*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

-((d + e*x)*(3*d^3*g - 2*e^3*f*x^2 + 3*d*e^2*x*(2*f + g*x) - d^2*e*(7*f + 9*g*x)))/(15*d^3*e^2*(d - e*x)^2*Sqr
t[d^2 - e^2*x^2])

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Maple [A]  time = 0.05, size = 85, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{4} \left ( 3\,d{e}^{2}g{x}^{2}-2\,{e}^{3}f{x}^{2}-9\,{d}^{2}egx+6\,d{e}^{2}fx+3\,{d}^{3}g-7\,{d}^{2}ef \right ) }{15\,{d}^{3}{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(g*x+f)/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/15*(-e*x+d)*(e*x+d)^4*(3*d*e^2*g*x^2-2*e^3*f*x^2-9*d^2*e*g*x+6*d*e^2*f*x+3*d^3*g-7*d^2*e*f)/d^3/e^2/(-e^2*x
^2+d^2)^(7/2)

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Maxima [B]  time = 1.02423, size = 504, normalized size = 4.31 \begin{align*} \frac{e g x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{d f x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{3 \, d^{2} g x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{3 \, d^{2} f}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d^{3} g}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} + \frac{4 \, f x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d} + \frac{g x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} + \frac{8 \, f x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}} + \frac{g x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} e} + \frac{{\left (e^{3} f + 3 \, d e^{2} g\right )} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} + \frac{3 \,{\left (d e^{2} f + d^{2} e g\right )} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{2 \,{\left (e^{3} f + 3 \, d e^{2} g\right )} d^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} - \frac{{\left (d e^{2} f + d^{2} e g\right )} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2} e^{2}} - \frac{2 \,{\left (d e^{2} f + d^{2} e g\right )} x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4} e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

1/2*e*g*x^3/(-e^2*x^2 + d^2)^(5/2) + 1/5*d*f*x/(-e^2*x^2 + d^2)^(5/2) - 3/10*d^2*g*x/((-e^2*x^2 + d^2)^(5/2)*e
) + 3/5*d^2*f/((-e^2*x^2 + d^2)^(5/2)*e) + 1/5*d^3*g/((-e^2*x^2 + d^2)^(5/2)*e^2) + 4/15*f*x/((-e^2*x^2 + d^2)
^(3/2)*d) + 1/10*g*x/((-e^2*x^2 + d^2)^(3/2)*e) + 8/15*f*x/(sqrt(-e^2*x^2 + d^2)*d^3) + 1/5*g*x/(sqrt(-e^2*x^2
+ d^2)*d^2*e) + 1/3*(e^3*f + 3*d*e^2*g)*x^2/((-e^2*x^2 + d^2)^(5/2)*e^2) + 3/5*(d*e^2*f + d^2*e*g)*x/((-e^2*x
^2 + d^2)^(5/2)*e^2) - 2/15*(e^3*f + 3*d*e^2*g)*d^2/((-e^2*x^2 + d^2)^(5/2)*e^4) - 1/5*(d*e^2*f + d^2*e*g)*x/(
(-e^2*x^2 + d^2)^(3/2)*d^2*e^2) - 2/5*(d*e^2*f + d^2*e*g)*x/(sqrt(-e^2*x^2 + d^2)*d^4*e^2)

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Fricas [A]  time = 1.89551, size = 378, normalized size = 3.23 \begin{align*} -\frac{7 \, d^{3} e f - 3 \, d^{4} g -{\left (7 \, e^{4} f - 3 \, d e^{3} g\right )} x^{3} + 3 \,{\left (7 \, d e^{3} f - 3 \, d^{2} e^{2} g\right )} x^{2} - 3 \,{\left (7 \, d^{2} e^{2} f - 3 \, d^{3} e g\right )} x +{\left (7 \, d^{2} e f - 3 \, d^{3} g +{\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} x^{2} - 3 \,{\left (2 \, d e^{2} f - 3 \, d^{2} e g\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{5} x^{3} - 3 \, d^{4} e^{4} x^{2} + 3 \, d^{5} e^{3} x - d^{6} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/15*(7*d^3*e*f - 3*d^4*g - (7*e^4*f - 3*d*e^3*g)*x^3 + 3*(7*d*e^3*f - 3*d^2*e^2*g)*x^2 - 3*(7*d^2*e^2*f - 3*
d^3*e*g)*x + (7*d^2*e*f - 3*d^3*g + (2*e^3*f - 3*d*e^2*g)*x^2 - 3*(2*d*e^2*f - 3*d^2*e*g)*x)*sqrt(-e^2*x^2 + d
^2))/(d^3*e^5*x^3 - 3*d^4*e^4*x^2 + 3*d^5*e^3*x - d^6*e^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{3} \left (f + g x\right )}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(g*x+f)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**3*(f + g*x)/(-(-d + e*x)*(d + e*x))**(7/2), x)

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Giac [A]  time = 1.23002, size = 188, normalized size = 1.61 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (15 \, d f -{\left (x{\left (\frac{{\left (3 \, d^{2} g e^{7} - 2 \, d f e^{8}\right )} x^{2} e^{\left (-4\right )}}{d^{4}} - \frac{5 \,{\left (3 \, d^{4} g e^{5} - d^{3} f e^{6}\right )} e^{\left (-4\right )}}{d^{4}}\right )} - \frac{5 \,{\left (3 \, d^{5} g e^{4} + d^{4} f e^{5}\right )} e^{\left (-4\right )}}{d^{4}}\right )} x\right )} x - \frac{{\left (3 \, d^{7} g e^{2} - 7 \, d^{6} f e^{3}\right )} e^{\left (-4\right )}}{d^{4}}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*((15*d*f - (x*((3*d^2*g*e^7 - 2*d*f*e^8)*x^2*e^(-4)/d^4 - 5*(3*d^4*g*e^5 - d^3*f*e^
6)*e^(-4)/d^4) - 5*(3*d^5*g*e^4 + d^4*f*e^5)*e^(-4)/d^4)*x)*x - (3*d^7*g*e^2 - 7*d^6*f*e^3)*e^(-4)/d^4)/(x^2*e
^2 - d^2)^3