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

Optimal. Leaf size=257 \[ \frac{16 g^2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{5 c^3 d^3 e}+\frac{12 g (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c^2 d^2 \sqrt{d+e x}}-\frac{16 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (2 a e^2 g-c d (3 e f-d g)\right )}{5 c^4 d^4 e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (f+g x)^3}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-2*Sqrt[d + e*x]*(f + g*x)^3)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (16*g*(c*d*f - a*e*g)*(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])/(5*c^4*d^4*e*Sqrt[d + e*x]) + (16*g^2*
(c*d*f - a*e*g)*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c^3*d^3*e) + (12*g*(f + g*x)^2*S
qrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c^2*d^2*Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 0.330761, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {866, 870, 794, 648} \[ \frac{16 g^2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{5 c^3 d^3 e}+\frac{12 g (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c^2 d^2 \sqrt{d+e x}}-\frac{16 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (2 a e^2 g-c d (3 e f-d g)\right )}{5 c^4 d^4 e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (f+g x)^3}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(f + g*x)^3)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (16*g*(c*d*f - a*e*g)*(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])/(5*c^4*d^4*e*Sqrt[d + e*x]) + (16*g^2*
(c*d*f - a*e*g)*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c^3*d^3*e) + (12*g*(f + g*x)^2*S
qrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c^2*d^2*Sqrt[d + e*x])

Rule 866

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*(p + 1)), x] - Dist[(e*g*n)/(c*(p + 1)), I
nt[(d + e*x)^(m - 1)*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, 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] &&  !IntegerQ[p] && EqQ[m + p, 0] &
& LtQ[p, -1] && GtQ[n, 0]

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{(d+e x)^{3/2} (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}} \, dx &=-\frac{2 \sqrt{d+e x} (f+g x)^3}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{(6 g) \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}{c d}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^3}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{12 g (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^2 d^2 \sqrt{d+e x}}+\frac{(24 g (c d f-a e g)) \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}{5 c^2 d^2}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^3}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{16 g^2 (c d f-a e g) \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^3 d^3 e}+\frac{12 g (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^2 d^2 \sqrt{d+e x}}-\frac{\left (8 g (c d f-a e g) \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}{5 c^3 d^3 e}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^3}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{16 g (c d f-a e g) \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}}{5 c^4 d^4 e \sqrt{d+e x}}+\frac{16 g^2 (c d f-a e g) \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^3 d^3 e}+\frac{12 g (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^2 d^2 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.111001, size = 134, normalized size = 0.52 \[ \frac{2 \sqrt{d+e x} \left (8 a^2 c d e^2 g^2 (g x-5 f)+16 a^3 e^3 g^3-2 a c^2 d^2 e g \left (-15 f^2+10 f g x+g^2 x^2\right )+c^3 d^3 \left (15 f^2 g x-5 f^3+5 f g^2 x^2+g^3 x^3\right )\right )}{5 c^4 d^4 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(16*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(-5*f + g*x) - 2*a*c^2*d^2*e*g*(-15*f^2 + 10*f*g*x + g^2*
x^2) + c^3*d^3*(-5*f^3 + 15*f^2*g*x + 5*f*g^2*x^2 + g^3*x^3)))/(5*c^4*d^4*Sqrt[(a*e + c*d*x)*(d + e*x)])

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 187, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ({g}^{3}{x}^{3}{c}^{3}{d}^{3}-2\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}+5\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}+8\,{a}^{2}cd{e}^{2}{g}^{3}x-20\,a{c}^{2}{d}^{2}ef{g}^{2}x+15\,{c}^{3}{d}^{3}{f}^{2}gx+16\,{a}^{3}{e}^{3}{g}^{3}-40\,{a}^{2}cd{e}^{2}f{g}^{2}+30\,a{c}^{2}{d}^{2}e{f}^{2}g-5\,{f}^{3}{c}^{3}{d}^{3} \right ) }{5\,{c}^{4}{d}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c*d*x+a*e)*(c^3*d^3*g^3*x^3-2*a*c^2*d^2*e*g^3*x^2+5*c^3*d^3*f*g^2*x^2+8*a^2*c*d*e^2*g^3*x-20*a*c^2*d^2*e*
f*g^2*x+15*c^3*d^3*f^2*g*x+16*a^3*e^3*g^3-40*a^2*c*d*e^2*f*g^2+30*a*c^2*d^2*e*f^2*g-5*c^3*d^3*f^3)*(e*x+d)^(3/
2)/c^4/d^4/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.2193, size = 223, normalized size = 0.87 \begin{align*} -\frac{2 \, f^{3}}{\sqrt{c d x + a e} c d} + \frac{6 \,{\left (c d x + 2 \, a e\right )} f^{2} g}{\sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (c^{2} d^{2} x^{2} - 4 \, a c d e x - 8 \, a^{2} e^{2}\right )} f g^{2}}{\sqrt{c d x + a e} c^{3} d^{3}} + \frac{2 \,{\left (c^{3} d^{3} x^{3} - 2 \, a c^{2} d^{2} e x^{2} + 8 \, a^{2} c d e^{2} x + 16 \, a^{3} e^{3}\right )} g^{3}}{5 \, \sqrt{c d x + a e} c^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*f^3/(sqrt(c*d*x + a*e)*c*d) + 6*(c*d*x + 2*a*e)*f^2*g/(sqrt(c*d*x + a*e)*c^2*d^2) + 2*(c^2*d^2*x^2 - 4*a*c*
d*e*x - 8*a^2*e^2)*f*g^2/(sqrt(c*d*x + a*e)*c^3*d^3) + 2/5*(c^3*d^3*x^3 - 2*a*c^2*d^2*e*x^2 + 8*a^2*c*d*e^2*x
+ 16*a^3*e^3)*g^3/(sqrt(c*d*x + a*e)*c^4*d^4)

________________________________________________________________________________________

Fricas [A]  time = 1.6162, size = 446, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (c^{3} d^{3} g^{3} x^{3} - 5 \, c^{3} d^{3} f^{3} + 30 \, a c^{2} d^{2} e f^{2} g - 40 \, a^{2} c d e^{2} f g^{2} + 16 \, a^{3} e^{3} g^{3} +{\left (5 \, c^{3} d^{3} f g^{2} - 2 \, a c^{2} d^{2} e g^{3}\right )} x^{2} +{\left (15 \, c^{3} d^{3} f^{2} g - 20 \, 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}}{5 \,{\left (c^{5} d^{5} e x^{2} + a c^{4} d^{5} e +{\left (c^{5} d^{6} + a c^{4} d^{4} e^{2}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c^3*d^3*g^3*x^3 - 5*c^3*d^3*f^3 + 30*a*c^2*d^2*e*f^2*g - 40*a^2*c*d*e^2*f*g^2 + 16*a^3*e^3*g^3 + (5*c^3*d
^3*f*g^2 - 2*a*c^2*d^2*e*g^3)*x^2 + (15*c^3*d^3*f^2*g - 20*a*c^2*d^2*e*f*g^2 + 8*a^2*c*d*e^2*g^3)*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^2 + a*c^4*d^5*e + (c^5*d^6 + a*c^4*d^4*e^2)*x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x