3.691 \(\int \frac{(f+g x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=200 \[ \frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{63 c^2 d^2 e (d+e x)^{3/2}}-\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{315 c^3 d^3 e (d+e x)^{5/2}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}} \]

[Out]

(-8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(7*e*f - 5*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*c^3*
d^3*e*(d + e*x)^(5/2)) + (8*g*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(63*c^2*d^2*e*(d
+ e*x)^(3/2)) + (2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*c*d*(d + e*x)^(5/2))

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Rubi [A]  time = 0.232528, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {870, 794, 648} \[ \frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{63 c^2 d^2 e (d+e x)^{3/2}}-\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{315 c^3 d^3 e (d+e x)^{5/2}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(7*e*f - 5*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*c^3*
d^3*e*(d + e*x)^(5/2)) + (8*g*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(63*c^2*d^2*e*(d
+ e*x)^(3/2)) + (2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*c*d*(d + e*x)^(5/2))

Rule 870

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
-Simp[(e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1))/(c*(m - n - 1)), x] - Dist[(n*(c*e*f + c*d*g
- b*e*g))/(c*e*(m - n - 1)), Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Integ
erQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int \frac{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=\frac{2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}}+\frac{\left (4 \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\right ) \int \frac{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{9 c d e^2}\\ &=\frac{8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 c^2 d^2 e (d+e x)^{3/2}}+\frac{2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}}-\frac{\left (4 (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right )\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{63 c^2 d^2 e}\\ &=-\frac{8 (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 c^3 d^3 e (d+e x)^{5/2}}+\frac{8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 c^2 d^2 e (d+e x)^{3/2}}+\frac{2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.124801, size = 90, normalized size = 0.45 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (8 a^2 e^2 g^2-4 a c d e g (9 f+5 g x)+c^2 d^2 \left (63 f^2+90 f g x+35 g^2 x^2\right )\right )}{315 c^3 d^3 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(9*f + 5*g*x) + c^2*d^2*(63*f^2 + 90*f*g*x + 3
5*g^2*x^2)))/(315*c^3*d^3*(d + e*x)^(5/2))

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Maple [A]  time = 0.05, size = 116, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 35\,{g}^{2}{x}^{2}{c}^{2}{d}^{2}-20\,acde{g}^{2}x+90\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-36\,acdefg+63\,{f}^{2}{c}^{2}{d}^{2} \right ) }{315\,{c}^{3}{d}^{3}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x)

[Out]

2/315*(c*d*x+a*e)*(35*c^2*d^2*g^2*x^2-20*a*c*d*e*g^2*x+90*c^2*d^2*f*g*x+8*a^2*e^2*g^2-36*a*c*d*e*f*g+63*c^2*d^
2*f^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)/c^3/d^3/(e*x+d)^(3/2)

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Maxima [A]  time = 1.18828, size = 259, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d x + a e} f^{2}}{5 \, c d} + \frac{4 \,{\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} f g}{35 \, c^{2} d^{2}} + \frac{2 \,{\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} e x^{3} + 3 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 4 \, a^{3} c d e^{3} x + 8 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} g^{2}}{315 \, c^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2)*sqrt(c*d*x + a*e)*f^2/(c*d) + 4/35*(5*c^3*d^3*x^3 + 8*a*c^2*d^2*e*x^
2 + a^2*c*d*e^2*x - 2*a^3*e^3)*sqrt(c*d*x + a*e)*f*g/(c^2*d^2) + 2/315*(35*c^4*d^4*x^4 + 50*a*c^3*d^3*e*x^3 +
3*a^2*c^2*d^2*e^2*x^2 - 4*a^3*c*d*e^3*x + 8*a^4*e^4)*sqrt(c*d*x + a*e)*g^2/(c^3*d^3)

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Fricas [A]  time = 1.61764, size = 481, normalized size = 2.4 \begin{align*} \frac{2 \,{\left (35 \, c^{4} d^{4} g^{2} x^{4} + 63 \, a^{2} c^{2} d^{2} e^{2} f^{2} - 36 \, a^{3} c d e^{3} f g + 8 \, a^{4} e^{4} g^{2} + 10 \,{\left (9 \, c^{4} d^{4} f g + 5 \, a c^{3} d^{3} e g^{2}\right )} x^{3} + 3 \,{\left (21 \, c^{4} d^{4} f^{2} + 48 \, a c^{3} d^{3} e f g + a^{2} c^{2} d^{2} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (63 \, a c^{3} d^{3} e f^{2} + 9 \, a^{2} c^{2} d^{2} e^{2} f g - 2 \, a^{3} c d e^{3} g^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{315 \,{\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*c^4*d^4*g^2*x^4 + 63*a^2*c^2*d^2*e^2*f^2 - 36*a^3*c*d*e^3*f*g + 8*a^4*e^4*g^2 + 10*(9*c^4*d^4*f*g +
5*a*c^3*d^3*e*g^2)*x^3 + 3*(21*c^4*d^4*f^2 + 48*a*c^3*d^3*e*f*g + a^2*c^2*d^2*e^2*g^2)*x^2 + 2*(63*a*c^3*d^3*e
*f^2 + 9*a^2*c^2*d^2*e^2*f*g - 2*a^3*c*d*e^3*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)
/(c^3*d^3*e*x + c^3*d^4)

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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((g*x+f)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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Giac [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((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

Timed out