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

Optimal. Leaf size=269 \[ \frac{12 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{35 c^2 d^2 \sqrt{d+e x}}+\frac{16 g \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{35 c^3 d^3 e}-\frac{16 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (2 a e^2 g-c d (3 e f-d g)\right )}{35 c^4 d^4 e \sqrt{d+e x}}+\frac{2 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt{d+e x}} \]

[Out]

(-16*(c*d*f - a*e*g)^2*(2*a*e^2*g - c*d*(3*e*f - d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^4*d^
4*e*Sqrt[d + e*x]) + (16*g*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^
3*d^3*e) + (12*(c*d*f - a*e*g)*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^2*d^2*Sqrt[d + e
*x]) + (2*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*c*d*Sqrt[d + e*x])

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Rubi [A]  time = 0.423284, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {870, 794, 648} \[ \frac{12 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{35 c^2 d^2 \sqrt{d+e x}}+\frac{16 g \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{35 c^3 d^3 e}-\frac{16 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (2 a e^2 g-c d (3 e f-d g)\right )}{35 c^4 d^4 e \sqrt{d+e x}}+\frac{2 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*(c*d*f - a*e*g)^2*(2*a*e^2*g - c*d*(3*e*f - d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^4*d^
4*e*Sqrt[d + e*x]) + (16*g*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^
3*d^3*e) + (12*(c*d*f - a*e*g)*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^2*d^2*Sqrt[d + e
*x]) + (2*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*c*d*Sqrt[d + e*x])

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{\sqrt{d+e x} (f+g x)^3}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d \sqrt{d+e x}}+\frac{\left (6 \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{\sqrt{d+e x} (f+g x)^2}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 c d e^2}\\ &=\frac{12 (c d f-a e g) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d \sqrt{d+e x}}+\frac{\left (24 (c d f-a e g)^2\right ) \int \frac{\sqrt{d+e x} (f+g x)}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 c^2 d^2}\\ &=\frac{16 g (c d f-a e g)^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3 e}+\frac{12 (c d f-a e g) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d \sqrt{d+e x}}-\frac{\left (8 (c d f-a e g)^2 \left (2 a e^2 g-c d (3 e f-d g)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 c^3 d^3 e}\\ &=-\frac{16 (c d f-a e g)^2 \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^4 d^4 e \sqrt{d+e x}}+\frac{16 g (c d f-a e g)^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3 e}+\frac{12 (c d f-a e g) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.13536, size = 136, normalized size = 0.51 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 c d e^2 g^2 (7 f+g x)-16 a^3 e^3 g^3-2 a c^2 d^2 e g \left (35 f^2+14 f g x+3 g^2 x^2\right )+c^3 d^3 \left (35 f^2 g x+35 f^3+21 f g^2 x^2+5 g^3 x^3\right )\right )}{35 c^4 d^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(7*f + g*x) - 2*a*c^2*d^2*e*g*(35*f^2 +
14*f*g*x + 3*g^2*x^2) + c^3*d^3*(35*f^3 + 35*f^2*g*x + 21*f*g^2*x^2 + 5*g^3*x^3)))/(35*c^4*d^4*Sqrt[d + e*x])

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Maple [A]  time = 0.051, size = 188, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -5\,{g}^{3}{x}^{3}{c}^{3}{d}^{3}+6\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-21\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-8\,{a}^{2}cd{e}^{2}{g}^{3}x+28\,a{c}^{2}{d}^{2}ef{g}^{2}x-35\,{c}^{3}{d}^{3}{f}^{2}gx+16\,{a}^{3}{e}^{3}{g}^{3}-56\,{a}^{2}cd{e}^{2}f{g}^{2}+70\,a{c}^{2}{d}^{2}e{f}^{2}g-35\,{f}^{3}{c}^{3}{d}^{3} \right ) }{35\,{c}^{4}{d}^{4}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-2/35*(c*d*x+a*e)*(-5*c^3*d^3*g^3*x^3+6*a*c^2*d^2*e*g^3*x^2-21*c^3*d^3*f*g^2*x^2-8*a^2*c*d*e^2*g^3*x+28*a*c^2*
d^2*e*f*g^2*x-35*c^3*d^3*f^2*g*x+16*a^3*e^3*g^3-56*a^2*c*d*e^2*f*g^2+70*a*c^2*d^2*e*f^2*g-35*c^3*d^3*f^3)*(e*x
+d)^(1/2)/c^4/d^4/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

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Maxima [A]  time = 1.21868, size = 294, normalized size = 1.09 \begin{align*} \frac{2 \, \sqrt{c d x + a e} f^{3}}{c d} + \frac{2 \,{\left (c^{2} d^{2} x^{2} - a c d e x - 2 \, a^{2} e^{2}\right )} f^{2} g}{\sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (3 \, c^{3} d^{3} x^{3} - a c^{2} d^{2} e x^{2} + 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} f g^{2}}{5 \, \sqrt{c d x + a e} c^{3} d^{3}} + \frac{2 \,{\left (5 \, c^{4} d^{4} x^{4} - a c^{3} d^{3} e x^{3} + 2 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} g^{3}}{35 \, \sqrt{c d x + a e} c^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(c*d*x + a*e)*f^3/(c*d) + 2*(c^2*d^2*x^2 - a*c*d*e*x - 2*a^2*e^2)*f^2*g/(sqrt(c*d*x + a*e)*c^2*d^2) + 2/
5*(3*c^3*d^3*x^3 - a*c^2*d^2*e*x^2 + 4*a^2*c*d*e^2*x + 8*a^3*e^3)*f*g^2/(sqrt(c*d*x + a*e)*c^3*d^3) + 2/35*(5*
c^4*d^4*x^4 - a*c^3*d^3*e*x^3 + 2*a^2*c^2*d^2*e^2*x^2 - 8*a^3*c*d*e^3*x - 16*a^4*e^4)*g^3/(sqrt(c*d*x + a*e)*c
^4*d^4)

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Fricas [A]  time = 1.7912, size = 405, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} g^{3} x^{3} + 35 \, c^{3} d^{3} f^{3} - 70 \, a c^{2} d^{2} e f^{2} g + 56 \, a^{2} c d e^{2} f g^{2} - 16 \, a^{3} e^{3} g^{3} + 3 \,{\left (7 \, c^{3} d^{3} f g^{2} - 2 \, a c^{2} d^{2} e g^{3}\right )} x^{2} +{\left (35 \, c^{3} d^{3} f^{2} g - 28 \, a c^{2} d^{2} e f g^{2} + 8 \, a^{2} c d e^{2} g^{3}\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}}{35 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*d^3*g^3*x^3 + 35*c^3*d^3*f^3 - 70*a*c^2*d^2*e*f^2*g + 56*a^2*c*d*e^2*f*g^2 - 16*a^3*e^3*g^3 + 3*(7
*c^3*d^3*f*g^2 - 2*a*c^2*d^2*e*g^3)*x^2 + (35*c^3*d^3*f^2*g - 28*a*c^2*d^2*e*f*g^2 + 8*a^2*c*d*e^2*g^3)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^4*d^4*e*x + c^4*d^5)

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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)**3*(e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

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