### 3.1993 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{\sqrt{d+e x}} \, dx$$

Optimal. Leaf size=119 $-\frac{6 c^2 d^2 (d+e x)^{11/2} \left (c d^2-a e^2\right )}{11 e^4}+\frac{2 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )^2}{3 e^4}-\frac{2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^3}{7 e^4}+\frac{2 c^3 d^3 (d+e x)^{13/2}}{13 e^4}$

[Out]

(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(7/2))/(7*e^4) + (2*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(9/2))/(3*e^4) - (6*c^2*d^
2*(c*d^2 - a*e^2)*(d + e*x)^(11/2))/(11*e^4) + (2*c^3*d^3*(d + e*x)^(13/2))/(13*e^4)

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Rubi [A]  time = 0.0569964, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.054, Rules used = {626, 43} $-\frac{6 c^2 d^2 (d+e x)^{11/2} \left (c d^2-a e^2\right )}{11 e^4}+\frac{2 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )^2}{3 e^4}-\frac{2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^3}{7 e^4}+\frac{2 c^3 d^3 (d+e x)^{13/2}}{13 e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(7/2))/(7*e^4) + (2*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(9/2))/(3*e^4) - (6*c^2*d^
2*(c*d^2 - a*e^2)*(d + e*x)^(11/2))/(11*e^4) + (2*c^3*d^3*(d + e*x)^(13/2))/(13*e^4)

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{\sqrt{d+e x}} \, dx &=\int (a e+c d x)^3 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3 (d+e x)^{5/2}}{e^3}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{7/2}}{e^3}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{9/2}}{e^3}+\frac{c^3 d^3 (d+e x)^{11/2}}{e^3}\right ) \, dx\\ &=-\frac{2 \left (c d^2-a e^2\right )^3 (d+e x)^{7/2}}{7 e^4}+\frac{2 c d \left (c d^2-a e^2\right )^2 (d+e x)^{9/2}}{3 e^4}-\frac{6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{11/2}}{11 e^4}+\frac{2 c^3 d^3 (d+e x)^{13/2}}{13 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0779152, size = 98, normalized size = 0.82 $\frac{2 (d+e x)^{7/2} \left (-819 c^2 d^2 (d+e x)^2 \left (c d^2-a e^2\right )+1001 c d (d+e x) \left (c d^2-a e^2\right )^2-429 \left (c d^2-a e^2\right )^3+231 c^3 d^3 (d+e x)^3\right )}{3003 e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(-429*(c*d^2 - a*e^2)^3 + 1001*c*d*(c*d^2 - a*e^2)^2*(d + e*x) - 819*c^2*d^2*(c*d^2 - a*e^2
)*(d + e*x)^2 + 231*c^3*d^3*(d + e*x)^3))/(3003*e^4)

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Maple [A]  time = 0.045, size = 131, normalized size = 1.1 \begin{align*}{\frac{462\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+1638\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-252\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+2002\,{a}^{2}cd{e}^{5}x-728\,a{c}^{2}{d}^{3}{e}^{3}x+112\,{c}^{3}{d}^{5}ex+858\,{a}^{3}{e}^{6}-572\,{a}^{2}c{d}^{2}{e}^{4}+208\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{3003\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(e*x+d)^(7/2)*(231*c^3*d^3*e^3*x^3+819*a*c^2*d^2*e^4*x^2-126*c^3*d^4*e^2*x^2+1001*a^2*c*d*e^5*x-364*a*c
^2*d^3*e^3*x+56*c^3*d^5*e*x+429*a^3*e^6-286*a^2*c*d^2*e^4+104*a*c^2*d^4*e^2-16*c^3*d^6)/e^4

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Maxima [B]  time = 1.01659, size = 825, normalized size = 6.93 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(15015*sqrt(e*x + d)*a^3*d^3*e^3 + 3003*((3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*
d^2)*c*d/e + 5*(c*d^2 + a*e^2)*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)/e)*a^2*d^2*e^2 + 5*(231*(e*x + d)^(13/2)
- 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6
006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3*d^3/e^3 + 143*((35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)
*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2*d^2/e^2 + 18*(5*(e*x + d)^
(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*(c*d^2 + a*e^2)*c*d/e^2 + 21*(c*
d^2 + a*e^2)^2*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)/e^2)*a*d*e + 65*(63*(e*x + d)
^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^
4 - 693*sqrt(e*x + d)*d^5)*(c*d^2 + a*e^2)*c^2*d^2/e^3 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378
*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*(c*d^2 + a*e^2)^2*c*d/e^3 + 429*(5*(e*
x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*(c*d^2 + a*e^2)^3/e^3)/e

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Fricas [B]  time = 1.91656, size = 620, normalized size = 5.21 \begin{align*} \frac{2 \,{\left (231 \, c^{3} d^{3} e^{6} x^{6} - 16 \, c^{3} d^{9} + 104 \, a c^{2} d^{7} e^{2} - 286 \, a^{2} c d^{5} e^{4} + 429 \, a^{3} d^{3} e^{6} + 63 \,{\left (9 \, c^{3} d^{4} e^{5} + 13 \, a c^{2} d^{2} e^{7}\right )} x^{5} + 7 \,{\left (53 \, c^{3} d^{5} e^{4} + 299 \, a c^{2} d^{3} e^{6} + 143 \, a^{2} c d e^{8}\right )} x^{4} +{\left (5 \, c^{3} d^{6} e^{3} + 1469 \, a c^{2} d^{4} e^{5} + 2717 \, a^{2} c d^{2} e^{7} + 429 \, a^{3} e^{9}\right )} x^{3} - 3 \,{\left (2 \, c^{3} d^{7} e^{2} - 13 \, a c^{2} d^{5} e^{4} - 715 \, a^{2} c d^{3} e^{6} - 429 \, a^{3} d e^{8}\right )} x^{2} +{\left (8 \, c^{3} d^{8} e - 52 \, a c^{2} d^{6} e^{3} + 143 \, a^{2} c d^{4} e^{5} + 1287 \, a^{3} d^{2} e^{7}\right )} x\right )} \sqrt{e x + d}}{3003 \, e^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(231*c^3*d^3*e^6*x^6 - 16*c^3*d^9 + 104*a*c^2*d^7*e^2 - 286*a^2*c*d^5*e^4 + 429*a^3*d^3*e^6 + 63*(9*c^3
*d^4*e^5 + 13*a*c^2*d^2*e^7)*x^5 + 7*(53*c^3*d^5*e^4 + 299*a*c^2*d^3*e^6 + 143*a^2*c*d*e^8)*x^4 + (5*c^3*d^6*e
^3 + 1469*a*c^2*d^4*e^5 + 2717*a^2*c*d^2*e^7 + 429*a^3*e^9)*x^3 - 3*(2*c^3*d^7*e^2 - 13*a*c^2*d^5*e^4 - 715*a^
2*c*d^3*e^6 - 429*a^3*d*e^8)*x^2 + (8*c^3*d^8*e - 52*a*c^2*d^6*e^3 + 143*a^2*c*d^4*e^5 + 1287*a^3*d^2*e^7)*x)*
sqrt(e*x + d)/e^4

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

[Out]

Timed out

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Giac [B]  time = 1.23625, size = 1254, normalized size = 10.54 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(429*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*c^3*d^
6*e^(-3) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c^2*d^5*e^(-1) + 143*(35*(
x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)
*d^4)*c^3*d^5*e^(-3) + 15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*c*d^4*e + 3861*(5*(x*e + d)^(7/2) - 21*
(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*c^2*d^4*e^(-1) + 65*(63*(x*e + d)^(11/2)
- 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*
sqrt(x*e + d)*d^5)*c^3*d^4*e^(-3) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2
*c*d^3*e + 429*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3
+ 315*sqrt(x*e + d)*d^4)*a*c^2*d^3*e^(-1) + 5*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d
)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e +
d)*d^6)*c^3*d^3*e^(-3) + 15015*sqrt(x*e + d)*a^3*d^3*e^3 + 15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*d^
2*e^3 + 3861*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*c*
d^2*e + 65*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 +
1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*c^2*d^2*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(
3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*d*e^3 + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/
2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*c*d*e + 429*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5
/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^3*e^3)*e^(-1)