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

Optimal. Leaf size=269 \[ -\frac {16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{315 c^4 d^4 e (d+e x)^{3/2}}+\frac {16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{105 c^3 d^3 e \sqrt {d+e x}}+\frac {4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}} \]

[Out]

-16/315*(-a*e*g+c*d*f)^2*(2*a*e^2*g-c*d*(-3*d*g+5*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^4/d^4/e/(e*x
+d)^(3/2)+4/21*(-a*e*g+c*d*f)*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/(e*x+d)^(3/2)+2/9*(g*x
+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/(e*x+d)^(3/2)+16/105*g*(-a*e*g+c*d*f)^2*(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(3/2)/c^3/d^3/e/(e*x+d)^(1/2)

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Rubi [A]  time = 0.39, antiderivative size = 269, normalized size of antiderivative = 1.00, 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} \[ \frac {4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{105 c^3 d^3 e \sqrt {d+e x}}-\frac {16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{315 c^4 d^4 e (d+e x)^{3/2}}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*(c*d*f - a*e*g)^2*(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))/(315*c
^4*d^4*e*(d + e*x)^(3/2)) + (16*g*(c*d*f - a*e*g)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*c^3*d^
3*e*Sqrt[d + e*x]) + (4*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(21*c^2*d^2
*(d + e*x)^(3/2)) + (2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*c*d*(d + e*x)^(3/2))

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]

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 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])

Rubi steps

\begin {align*} \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 &=\frac {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}}{9 c d (d+e x)^{3/2}}+\frac {\left (2 \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)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{3 c d e^2}\\ &=\frac {4 (c d f-a e g) (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}}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {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}}{9 c d (d+e x)^{3/2}}+\frac {\left (8 (c d f-a e g)^2\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}{21 c^2 d^2}\\ &=\frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 e \sqrt {d+e x}}+\frac {4 (c d f-a e g) (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}}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {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}}{9 c d (d+e x)^{3/2}}+\frac {\left (8 (c d f-a e g)^2 \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}{105 c^2 d^2}\\ &=\frac {16 (c d f-a e g)^2 \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}}{315 c^3 d^3 (d+e x)^{3/2}}+\frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 e \sqrt {d+e x}}+\frac {4 (c d f-a e g) (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}}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {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}}{9 c d (d+e x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 136, normalized size = 0.51 \[ \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (3 f+g x)-6 a c^2 d^2 e g \left (21 f^2+18 f g x+5 g^2 x^2\right )+c^3 d^3 \left (105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right )\right )}{315 c^4 d^4 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^3*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)*(-16*a^3*e^3*g^3 + 24*a^2*c*d*e^2*g^2*(3*f + g*x) - 6*a*c^2*d^2*e*g*(21*f^2
+ 18*f*g*x + 5*g^2*x^2) + c^3*d^3*(105*f^3 + 189*f^2*g*x + 135*f*g^2*x^2 + 35*g^3*x^3)))/(315*c^4*d^4*(d + e*
x)^(3/2))

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fricas [A]  time = 1.00, size = 264, normalized size = 0.98 \[ \frac {2 \, {\left (35 \, c^{4} d^{4} g^{3} x^{4} + 105 \, a c^{3} d^{3} e f^{3} - 126 \, a^{2} c^{2} d^{2} e^{2} f^{2} g + 72 \, a^{3} c d e^{3} f g^{2} - 16 \, a^{4} e^{4} g^{3} + 5 \, {\left (27 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{4} f^{2} g + 9 \, a c^{3} d^{3} e f g^{2} - 2 \, a^{2} c^{2} d^{2} e^{2} g^{3}\right )} x^{2} + {\left (105 \, c^{4} d^{4} f^{3} + 63 \, a c^{3} d^{3} e f^{2} g - 36 \, a^{2} c^{2} d^{2} e^{2} f g^{2} + 8 \, a^{3} c d e^{3} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{315 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(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]

2/315*(35*c^4*d^4*g^3*x^4 + 105*a*c^3*d^3*e*f^3 - 126*a^2*c^2*d^2*e^2*f^2*g + 72*a^3*c*d*e^3*f*g^2 - 16*a^4*e^
4*g^3 + 5*(27*c^4*d^4*f*g^2 + a*c^3*d^3*e*g^3)*x^3 + 3*(63*c^4*d^4*f^2*g + 9*a*c^3*d^3*e*f*g^2 - 2*a^2*c^2*d^2
*e^2*g^3)*x^2 + (105*c^4*d^4*f^3 + 63*a*c^3*d^3*e*f^2*g - 36*a^2*c^2*d^2*e^2*f*g^2 + 8*a^3*c*d*e^3*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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{3}}{\sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(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]

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

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maple [A]  time = 0.01, size = 188, normalized size = 0.70 \[ -\frac {2 \left (c d x +a e \right ) \left (-35 g^{3} x^{3} c^{3} d^{3}+30 a \,c^{2} d^{2} e \,g^{3} x^{2}-135 c^{3} d^{3} f \,g^{2} x^{2}-24 a^{2} c d \,e^{2} g^{3} x +108 a \,c^{2} d^{2} e f \,g^{2} x -189 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-72 a^{2} c d \,e^{2} f \,g^{2}+126 a \,c^{2} d^{2} e \,f^{2} g -105 f^{3} c^{3} d^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \sqrt {e x +d}\, c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(c*d*x+a*e)*(-35*c^3*d^3*g^3*x^3+30*a*c^2*d^2*e*g^3*x^2-135*c^3*d^3*f*g^2*x^2-24*a^2*c*d*e^2*g^3*x+108*
a*c^2*d^2*e*f*g^2*x-189*c^3*d^3*f^2*g*x+16*a^3*e^3*g^3-72*a^2*c*d*e^2*f*g^2+126*a*c^2*d^2*e*f^2*g-105*c^3*d^3*
f^3)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/c^4/d^4/(e*x+d)^(1/2)

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maxima [A]  time = 0.65, size = 218, normalized size = 0.81 \[ \frac {2 \, {\left (c d x + a e\right )}^{\frac {3}{2}} f^{3}}{3 \, c d} + \frac {2 \, {\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{2} g}{5 \, c^{2} d^{2}} + \frac {2 \, {\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f g^{2}}{35 \, c^{3} d^{3}} + \frac {2 \, {\left (35 \, c^{4} d^{4} x^{4} + 5 \, a c^{3} d^{3} e x^{3} - 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} g^{3}}{315 \, c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 3.37, size = 242, normalized size = 0.90 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^3\,x^4}{9}-\frac {32\,a^4\,e^4\,g^3-144\,a^3\,c\,d\,e^3\,f\,g^2+252\,a^2\,c^2\,d^2\,e^2\,f^2\,g-210\,a\,c^3\,d^3\,e\,f^3}{315\,c^4\,d^4}+\frac {x\,\left (16\,a^3\,c\,d\,e^3\,g^3-72\,a^2\,c^2\,d^2\,e^2\,f\,g^2+126\,a\,c^3\,d^3\,e\,f^2\,g+210\,c^4\,d^4\,f^3\right )}{315\,c^4\,d^4}+\frac {2\,g\,x^2\,\left (-2\,a^2\,e^2\,g^2+9\,a\,c\,d\,e\,f\,g+63\,c^2\,d^2\,f^2\right )}{105\,c^2\,d^2}+\frac {2\,g^2\,x^3\,\left (a\,e\,g+27\,c\,d\,f\right )}{63\,c\,d}\right )}{\sqrt {d+e\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*g^3*x^4)/9 - (32*a^4*e^4*g^3 - 210*a*c^3*d^3*e*f^3 + 252*a^
2*c^2*d^2*e^2*f^2*g - 144*a^3*c*d*e^3*f*g^2)/(315*c^4*d^4) + (x*(210*c^4*d^4*f^3 + 16*a^3*c*d*e^3*g^3 - 72*a^2
*c^2*d^2*e^2*f*g^2 + 126*a*c^3*d^3*e*f^2*g))/(315*c^4*d^4) + (2*g*x^2*(63*c^2*d^2*f^2 - 2*a^2*e^2*g^2 + 9*a*c*
d*e*f*g))/(105*c^2*d^2) + (2*g^2*x^3*(a*e*g + 27*c*d*f))/(63*c*d)))/(d + e*x)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{3}}{\sqrt {d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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