3.679 \(\int \frac{(f+g x)^4 \sqrt{a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=336 \[ \frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{99 c^2 d^2 (d+e x)^{3/2}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3}{1155 c^4 d^4 e \sqrt{d+e x}}-\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{3465 c^5 d^5 e (d+e x)^{3/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}} \]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3465
*c^5*d^5*e*(d + e*x)^(3/2)) + (128*g*(c*d*f - a*e*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(1155*c^
4*d^4*e*Sqrt[d + e*x]) + (32*(c*d*f - a*e*g)^2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(231
*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d*f - a*e*g)*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(99
*c^2*d^2*(d + e*x)^(3/2)) + (2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d*(d + e*x)^(3
/2))

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Rubi [A]  time = 0.607251, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {870, 794, 648} \[ \frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{99 c^2 d^2 (d+e x)^{3/2}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3}{1155 c^4 d^4 e \sqrt{d+e x}}-\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{3465 c^5 d^5 e (d+e x)^{3/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3465
*c^5*d^5*e*(d + e*x)^(3/2)) + (128*g*(c*d*f - a*e*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(1155*c^
4*d^4*e*Sqrt[d + e*x]) + (32*(c*d*f - a*e*g)^2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(231
*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d*f - a*e*g)*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(99
*c^2*d^2*(d + e*x)^(3/2)) + (2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d*(d + e*x)^(3
/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)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx &=\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac{(8 (c d f-a e g)) \int \frac{(f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{11 c d}\\ &=\frac{16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac{\left (16 (c d f-a e g)^2\right ) \int \frac{(f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{33 c^2 d^2}\\ &=\frac{32 (c d f-a e g)^2 (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}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac{\left (64 (c d f-a e g)^3\right ) \int \frac{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{231 c^3 d^3}\\ &=\frac{128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1155 c^4 d^4 e \sqrt{d+e x}}+\frac{32 (c d f-a e g)^2 (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}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac{\left (64 (c d f-a e g)^3 \left (5 f-\frac{3 d g}{e}-\frac{2 a e g}{c d}\right )\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{1155 c^3 d^3}\\ &=\frac{128 (c d f-a e g)^3 \left (5 f-\frac{3 d g}{e}-\frac{2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3465 c^4 d^4 (d+e x)^{3/2}}+\frac{128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1155 c^4 d^4 e \sqrt{d+e x}}+\frac{32 (c d f-a e g)^2 (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}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.185146, size = 195, normalized size = 0.58 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (48 a^2 c^2 d^2 e^2 g^2 \left (33 f^2+22 f g x+5 g^2 x^2\right )-64 a^3 c d e^3 g^3 (11 f+3 g x)+128 a^4 e^4 g^4-8 a c^3 d^3 e g \left (297 f^2 g x+231 f^3+165 f g^2 x^2+35 g^3 x^3\right )+c^4 d^4 \left (2970 f^2 g^2 x^2+2772 f^3 g x+1155 f^4+1540 f g^3 x^3+315 g^4 x^4\right )\right )}{3465 c^5 d^5 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*(11*f + 3*g*x) + 48*a^2*c^2*d^2*e^2*g
^2*(33*f^2 + 22*f*g*x + 5*g^2*x^2) - 8*a*c^3*d^3*e*g*(231*f^3 + 297*f^2*g*x + 165*f*g^2*x^2 + 35*g^3*x^3) + c^
4*d^4*(1155*f^4 + 2772*f^3*g*x + 2970*f^2*g^2*x^2 + 1540*f*g^3*x^3 + 315*g^4*x^4)))/(3465*c^5*d^5*(d + e*x)^(3
/2))

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Maple [A]  time = 0.054, size = 283, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 315\,{g}^{4}{x}^{4}{c}^{4}{d}^{4}-280\,a{c}^{3}{d}^{3}e{g}^{4}{x}^{3}+1540\,{c}^{4}{d}^{4}f{g}^{3}{x}^{3}+240\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{g}^{4}{x}^{2}-1320\,a{c}^{3}{d}^{3}ef{g}^{3}{x}^{2}+2970\,{c}^{4}{d}^{4}{f}^{2}{g}^{2}{x}^{2}-192\,{a}^{3}cd{e}^{3}{g}^{4}x+1056\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}f{g}^{3}x-2376\,a{c}^{3}{d}^{3}e{f}^{2}{g}^{2}x+2772\,{c}^{4}{d}^{4}{f}^{3}gx+128\,{a}^{4}{e}^{4}{g}^{4}-704\,{a}^{3}cd{e}^{3}f{g}^{3}+1584\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{f}^{2}{g}^{2}-1848\,a{c}^{3}{d}^{3}e{f}^{3}g+1155\,{f}^{4}{c}^{4}{d}^{4} \right ) }{3465\,{c}^{5}{d}^{5}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x)

[Out]

2/3465*(c*d*x+a*e)*(315*c^4*d^4*g^4*x^4-280*a*c^3*d^3*e*g^4*x^3+1540*c^4*d^4*f*g^3*x^3+240*a^2*c^2*d^2*e^2*g^4
*x^2-1320*a*c^3*d^3*e*f*g^3*x^2+2970*c^4*d^4*f^2*g^2*x^2-192*a^3*c*d*e^3*g^4*x+1056*a^2*c^2*d^2*e^2*f*g^3*x-23
76*a*c^3*d^3*e*f^2*g^2*x+2772*c^4*d^4*f^3*g*x+128*a^4*e^4*g^4-704*a^3*c*d*e^3*f*g^3+1584*a^2*c^2*d^2*e^2*f^2*g
^2-1848*a*c^3*d^3*e*f^3*g+1155*c^4*d^4*f^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/c^5/d^5/(e*x+d)^(1/2)

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Maxima [A]  time = 1.26439, size = 432, normalized size = 1.29 \begin{align*} \frac{2 \,{\left (c d x + a e\right )}^{\frac{3}{2}} f^{4}}{3 \, c d} + \frac{8 \,{\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt{c d x + a e} f^{3} g}{15 \, c^{2} d^{2}} + \frac{4 \,{\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} f^{2} g^{2}}{35 \, c^{3} d^{3}} + \frac{8 \,{\left (35 \, c^{4} d^{4} x^{4} + 5 \, a c^{3} d^{3} e x^{3} - 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} f g^{3}}{315 \, c^{4} d^{4}} + \frac{2 \,{\left (315 \, c^{5} d^{5} x^{5} + 35 \, a c^{4} d^{4} e x^{4} - 40 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 48 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 64 \, a^{4} c d e^{4} x + 128 \, a^{5} e^{5}\right )} \sqrt{c d x + a e} g^{4}}{3465 \, c^{5} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/3*(c*d*x + a*e)^(3/2)*f^4/(c*d) + 8/15*(3*c^2*d^2*x^2 + a*c*d*e*x - 2*a^2*e^2)*sqrt(c*d*x + a*e)*f^3*g/(c^2*
d^2) + 4/35*(15*c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 - 4*a^2*c*d*e^2*x + 8*a^3*e^3)*sqrt(c*d*x + a*e)*f^2*g^2/(c^3*
d^3) + 8/315*(35*c^4*d^4*x^4 + 5*a*c^3*d^3*e*x^3 - 6*a^2*c^2*d^2*e^2*x^2 + 8*a^3*c*d*e^3*x - 16*a^4*e^4)*sqrt(
c*d*x + a*e)*f*g^3/(c^4*d^4) + 2/3465*(315*c^5*d^5*x^5 + 35*a*c^4*d^4*e*x^4 - 40*a^2*c^3*d^3*e^2*x^3 + 48*a^3*
c^2*d^2*e^3*x^2 - 64*a^4*c*d*e^4*x + 128*a^5*e^5)*sqrt(c*d*x + a*e)*g^4/(c^5*d^5)

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Fricas [A]  time = 1.61815, size = 798, normalized size = 2.38 \begin{align*} \frac{2 \,{\left (315 \, c^{5} d^{5} g^{4} x^{5} + 1155 \, a c^{4} d^{4} e f^{4} - 1848 \, a^{2} c^{3} d^{3} e^{2} f^{3} g + 1584 \, a^{3} c^{2} d^{2} e^{3} f^{2} g^{2} - 704 \, a^{4} c d e^{4} f g^{3} + 128 \, a^{5} e^{5} g^{4} + 35 \,{\left (44 \, c^{5} d^{5} f g^{3} + a c^{4} d^{4} e g^{4}\right )} x^{4} + 10 \,{\left (297 \, c^{5} d^{5} f^{2} g^{2} + 22 \, a c^{4} d^{4} e f g^{3} - 4 \, a^{2} c^{3} d^{3} e^{2} g^{4}\right )} x^{3} + 6 \,{\left (462 \, c^{5} d^{5} f^{3} g + 99 \, a c^{4} d^{4} e f^{2} g^{2} - 44 \, a^{2} c^{3} d^{3} e^{2} f g^{3} + 8 \, a^{3} c^{2} d^{2} e^{3} g^{4}\right )} x^{2} +{\left (1155 \, c^{5} d^{5} f^{4} + 924 \, a c^{4} d^{4} e f^{3} g - 792 \, a^{2} c^{3} d^{3} e^{2} f^{2} g^{2} + 352 \, a^{3} c^{2} d^{2} e^{3} f g^{3} - 64 \, a^{4} c d e^{4} g^{4}\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}}{3465 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*c^5*d^5*g^4*x^5 + 1155*a*c^4*d^4*e*f^4 - 1848*a^2*c^3*d^3*e^2*f^3*g + 1584*a^3*c^2*d^2*e^3*f^2*g^2
- 704*a^4*c*d*e^4*f*g^3 + 128*a^5*e^5*g^4 + 35*(44*c^5*d^5*f*g^3 + a*c^4*d^4*e*g^4)*x^4 + 10*(297*c^5*d^5*f^2
*g^2 + 22*a*c^4*d^4*e*f*g^3 - 4*a^2*c^3*d^3*e^2*g^4)*x^3 + 6*(462*c^5*d^5*f^3*g + 99*a*c^4*d^4*e*f^2*g^2 - 44*
a^2*c^3*d^3*e^2*f*g^3 + 8*a^3*c^2*d^2*e^3*g^4)*x^2 + (1155*c^5*d^5*f^4 + 924*a*c^4*d^4*e*f^3*g - 792*a^2*c^3*d
^3*e^2*f^2*g^2 + 352*a^3*c^2*d^2*e^3*f*g^3 - 64*a^4*c*d*e^4*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
)*sqrt(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

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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)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(1/2),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)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

Timed out