3.2038 \(\int \sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\)

Optimal. Leaf size=171 \[ \frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 c^3 d^3 (d+e x)^{5/2}}+\frac {8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}} \]

[Out]

16/315*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3/(e*x+d)^(5/2)+8/63*(-a*e^2+c*d^2)*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^2/d^2/(e*x+d)^(3/2)+2/9*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/(e*x
+d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac {8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 c^2 d^2 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 c^3 d^3 (d+e x)^{5/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*c^3*d^3*(d + e*x)^(5/2)) + (8*(c*d^2
- a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(63*c^2*d^2*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(5/2))/(9*c*d*Sqrt[d + e*x])

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 656

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*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

\begin {align*} \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx &=\frac {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 \sqrt {d+e x}}+\frac {\left (4 \left (d^2-\frac {a e^2}{c}\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}}{\sqrt {d+e x}} \, dx}{9 d}\\ &=\frac {8 \left (c d^2-a e^2\right ) \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 (d+e x)^{3/2}}+\frac {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 \sqrt {d+e x}}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )^2\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 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \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 (d+e x)^{5/2}}+\frac {8 \left (c d^2-a e^2\right ) \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 (d+e x)^{3/2}}+\frac {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 \sqrt {d+e x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 88, normalized size = 0.51 \[ \frac {2 ((d+e x) (a e+c d x))^{5/2} \left (8 a^2 e^4-4 a c d e^2 (9 d+5 e x)+c^2 d^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )\right )}{315 c^3 d^3 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(8*a^2*e^4 - 4*a*c*d*e^2*(9*d + 5*e*x) + c^2*d^2*(63*d^2 + 90*d*e*x + 35*e^
2*x^2)))/(315*c^3*d^3*(d + e*x)^(5/2))

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 208, normalized size = 1.22 \[ \frac {2 \, {\left (35 \, c^{4} d^{4} e^{2} x^{4} + 63 \, a^{2} c^{2} d^{4} e^{2} - 36 \, a^{3} c d^{2} e^{4} + 8 \, a^{4} e^{6} + 10 \, {\left (9 \, c^{4} d^{5} e + 5 \, a c^{3} d^{3} e^{3}\right )} x^{3} + 3 \, {\left (21 \, c^{4} d^{6} + 48 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (63 \, a c^{3} d^{5} e + 9 \, a^{2} c^{2} d^{3} e^{3} - 2 \, a^{3} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{315 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*c^4*d^4*e^2*x^4 + 63*a^2*c^2*d^4*e^2 - 36*a^3*c*d^2*e^4 + 8*a^4*e^6 + 10*(9*c^4*d^5*e + 5*a*c^3*d^3*
e^3)*x^3 + 3*(21*c^4*d^6 + 48*a*c^3*d^4*e^2 + a^2*c^2*d^2*e^4)*x^2 + 2*(63*a*c^3*d^5*e + 9*a^2*c^2*d^3*e^3 - 2
*a^3*c*d*e^5)*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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} \sqrt {e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 110, normalized size = 0.64 \[ \frac {2 \left (c d x +a e \right ) \left (35 c^{2} d^{2} e^{2} x^{2}-20 a c d \,e^{3} x +90 c^{2} d^{3} e x +8 a^{2} e^{4}-36 a c \,d^{2} e^{2}+63 c^{2} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{\frac {3}{2}} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(c*d*x+a*e)*(35*c^2*d^2*e^2*x^2-20*a*c*d*e^3*x+90*c^2*d^3*e*x+8*a^2*e^4-36*a*c*d^2*e^2+63*c^2*d^4)*(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)

________________________________________________________________________________________

maxima [A]  time = 1.38, size = 189, normalized size = 1.11 \[ \frac {2 \, {\left (35 \, c^{4} d^{4} e^{2} x^{4} + 63 \, a^{2} c^{2} d^{4} e^{2} - 36 \, a^{3} c d^{2} e^{4} + 8 \, a^{4} e^{6} + 10 \, {\left (9 \, c^{4} d^{5} e + 5 \, a c^{3} d^{3} e^{3}\right )} x^{3} + 3 \, {\left (21 \, c^{4} d^{6} + 48 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (63 \, a c^{3} d^{5} e + 9 \, a^{2} c^{2} d^{3} e^{3} - 2 \, a^{3} c d e^{5}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{315 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*c^4*d^4*e^2*x^4 + 63*a^2*c^2*d^4*e^2 - 36*a^3*c*d^2*e^4 + 8*a^4*e^6 + 10*(9*c^4*d^5*e + 5*a*c^3*d^3*
e^3)*x^3 + 3*(21*c^4*d^6 + 48*a*c^3*d^4*e^2 + a^2*c^2*d^2*e^4)*x^2 + 2*(63*a*c^3*d^5*e + 9*a^2*c^2*d^3*e^3 - 2
*a^3*c*d*e^5)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^3*d^3*e*x + c^3*d^4)

________________________________________________________________________________________

mupad [B]  time = 1.02, size = 230, normalized size = 1.35 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (x^3\,\left (\frac {4\,c\,d^2}{7}+\frac {20\,a\,e^2}{63}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (16\,a^4\,e^6-72\,a^3\,c\,d^2\,e^4+126\,a^2\,c^2\,d^4\,e^2\right )}{315\,c^3\,d^3\,e}+\frac {2\,c\,d\,e\,x^4\,\sqrt {d+e\,x}}{9}+\frac {x^2\,\sqrt {d+e\,x}\,\left (6\,a^2\,c^2\,d^2\,e^4+288\,a\,c^3\,d^4\,e^2+126\,c^4\,d^6\right )}{315\,c^3\,d^3\,e}+\frac {4\,a\,x\,\sqrt {d+e\,x}\,\left (-2\,a^2\,e^4+9\,a\,c\,d^2\,e^2+63\,c^2\,d^4\right )}{315\,c^2\,d^2}\right )}{x+\frac {d}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x^3*((20*a*e^2)/63 + (4*c*d^2)/7)*(d + e*x)^(1/2) + ((d + e*x)
^(1/2)*(16*a^4*e^6 - 72*a^3*c*d^2*e^4 + 126*a^2*c^2*d^4*e^2))/(315*c^3*d^3*e) + (2*c*d*e*x^4*(d + e*x)^(1/2))/
9 + (x^2*(d + e*x)^(1/2)*(126*c^4*d^6 + 288*a*c^3*d^4*e^2 + 6*a^2*c^2*d^2*e^4))/(315*c^3*d^3*e) + (4*a*x*(d +
e*x)^(1/2)*(63*c^2*d^4 - 2*a^2*e^4 + 9*a*c*d^2*e^2))/(315*c^2*d^2)))/(x + d/e)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \sqrt {d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________