### 3.1992 $$\int \sqrt{d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx$$

Optimal. Leaf size=119 $-\frac{6 c^2 d^2 (d+e x)^{13/2} \left (c d^2-a e^2\right )}{13 e^4}+\frac{6 c d (d+e x)^{11/2} \left (c d^2-a e^2\right )^2}{11 e^4}-\frac{2 (d+e x)^{9/2} \left (c d^2-a e^2\right )^3}{9 e^4}+\frac{2 c^3 d^3 (d+e x)^{15/2}}{15 e^4}$

[Out]

(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(9/2))/(9*e^4) + (6*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(11/2))/(11*e^4) - (6*c^2*
d^2*(c*d^2 - a*e^2)*(d + e*x)^(13/2))/(13*e^4) + (2*c^3*d^3*(d + e*x)^(15/2))/(15*e^4)

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Rubi [A]  time = 0.0816524, 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} $-\frac{6 c^2 d^2 (d+e x)^{13/2} \left (c d^2-a e^2\right )}{13 e^4}+\frac{6 c d (d+e x)^{11/2} \left (c d^2-a e^2\right )^2}{11 e^4}-\frac{2 (d+e x)^{9/2} \left (c d^2-a e^2\right )^3}{9 e^4}+\frac{2 c^3 d^3 (d+e x)^{15/2}}{15 e^4}$

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,x]

[Out]

(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(9/2))/(9*e^4) + (6*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(11/2))/(11*e^4) - (6*c^2*
d^2*(c*d^2 - a*e^2)*(d + e*x)^(13/2))/(13*e^4) + (2*c^3*d^3*(d + e*x)^(15/2))/(15*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 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx &=\int (a e+c d x)^3 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3 (d+e x)^{7/2}}{e^3}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{9/2}}{e^3}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{11/2}}{e^3}+\frac{c^3 d^3 (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac{2 \left (c d^2-a e^2\right )^3 (d+e x)^{9/2}}{9 e^4}+\frac{6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{11/2}}{11 e^4}-\frac{6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{13/2}}{13 e^4}+\frac{2 c^3 d^3 (d+e x)^{15/2}}{15 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0915999, size = 111, normalized size = 0.93 $\frac{2 (d+e x)^{9/2} \left (-195 a^2 c d e^4 (2 d-9 e x)+715 a^3 e^6+15 a c^2 d^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+c^3 d^3 \left (72 d^2 e x-16 d^3-198 d e^2 x^2+429 e^3 x^3\right )\right )}{6435 e^4}$

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,x]

[Out]

(2*(d + e*x)^(9/2)*(715*a^3*e^6 - 195*a^2*c*d*e^4*(2*d - 9*e*x) + 15*a*c^2*d^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2*
x^2) + c^3*d^3*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3)))/(6435*e^4)

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Maple [A]  time = 0.046, size = 131, normalized size = 1.1 \begin{align*}{\frac{858\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+2970\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-396\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+3510\,{a}^{2}cd{e}^{5}x-1080\,a{c}^{2}{d}^{3}{e}^{3}x+144\,{c}^{3}{d}^{5}ex+1430\,{a}^{3}{e}^{6}-780\,{a}^{2}c{d}^{2}{e}^{4}+240\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{6435\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{9}{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]

2/6435*(e*x+d)^(9/2)*(429*c^3*d^3*e^3*x^3+1485*a*c^2*d^2*e^4*x^2-198*c^3*d^4*e^2*x^2+1755*a^2*c*d*e^5*x-540*a*
c^2*d^3*e^3*x+72*c^3*d^5*e*x+715*a^3*e^6-390*a^2*c*d^2*e^4+120*a*c^2*d^4*e^2-16*c^3*d^6)/e^4

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Maxima [A]  time = 1.03659, size = 185, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (429 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} d^{3} - 1485 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1755 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 715 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{6435 \, 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="maxima")

[Out]

2/6435*(429*(e*x + d)^(15/2)*c^3*d^3 - 1485*(c^3*d^4 - a*c^2*d^2*e^2)*(e*x + d)^(13/2) + 1755*(c^3*d^5 - 2*a*c
^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(11/2) - 715*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*(e*x
+ d)^(9/2))/e^4

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Fricas [B]  time = 1.91741, size = 743, normalized size = 6.24 \begin{align*} \frac{2 \,{\left (429 \, c^{3} d^{3} e^{7} x^{7} - 16 \, c^{3} d^{10} + 120 \, a c^{2} d^{8} e^{2} - 390 \, a^{2} c d^{6} e^{4} + 715 \, a^{3} d^{4} e^{6} + 33 \,{\left (46 \, c^{3} d^{4} e^{6} + 45 \, a c^{2} d^{2} e^{8}\right )} x^{6} + 9 \,{\left (206 \, c^{3} d^{5} e^{5} + 600 \, a c^{2} d^{3} e^{7} + 195 \, a^{2} c d e^{9}\right )} x^{5} + 5 \,{\left (160 \, c^{3} d^{6} e^{4} + 1374 \, a c^{2} d^{4} e^{6} + 1326 \, a^{2} c d^{2} e^{8} + 143 \, a^{3} e^{10}\right )} x^{4} + 5 \,{\left (c^{3} d^{7} e^{3} + 636 \, a c^{2} d^{5} e^{5} + 1794 \, a^{2} c d^{3} e^{7} + 572 \, a^{3} d e^{9}\right )} x^{3} - 3 \,{\left (2 \, c^{3} d^{8} e^{2} - 15 \, a c^{2} d^{6} e^{4} - 1560 \, a^{2} c d^{4} e^{6} - 1430 \, a^{3} d^{2} e^{8}\right )} x^{2} +{\left (8 \, c^{3} d^{9} e - 60 \, a c^{2} d^{7} e^{3} + 195 \, a^{2} c d^{5} e^{5} + 2860 \, a^{3} d^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{6435 \, 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]

2/6435*(429*c^3*d^3*e^7*x^7 - 16*c^3*d^10 + 120*a*c^2*d^8*e^2 - 390*a^2*c*d^6*e^4 + 715*a^3*d^4*e^6 + 33*(46*c
^3*d^4*e^6 + 45*a*c^2*d^2*e^8)*x^6 + 9*(206*c^3*d^5*e^5 + 600*a*c^2*d^3*e^7 + 195*a^2*c*d*e^9)*x^5 + 5*(160*c^
3*d^6*e^4 + 1374*a*c^2*d^4*e^6 + 1326*a^2*c*d^2*e^8 + 143*a^3*e^10)*x^4 + 5*(c^3*d^7*e^3 + 636*a*c^2*d^5*e^5 +
1794*a^2*c*d^3*e^7 + 572*a^3*d*e^9)*x^3 - 3*(2*c^3*d^8*e^2 - 15*a*c^2*d^6*e^4 - 1560*a^2*c*d^4*e^6 - 1430*a^3
*d^2*e^8)*x^2 + (8*c^3*d^9*e - 60*a*c^2*d^7*e^3 + 195*a^2*c*d^5*e^5 + 2860*a^3*d^3*e^7)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 7.85638, size = 165, normalized size = 1.39 \begin{align*} \frac{2 \left (\frac{c^{3} d^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{3}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (3 a c^{2} d^{2} e^{2} - 3 c^{3} d^{4}\right )}{13 e^{3}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (3 a^{2} c d e^{4} - 6 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}\right )}{11 e^{3}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}\right )}{9 e^{3}}\right )}{e} \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]

2*(c**3*d**3*(d + e*x)**(15/2)/(15*e**3) + (d + e*x)**(13/2)*(3*a*c**2*d**2*e**2 - 3*c**3*d**4)/(13*e**3) + (d
+ e*x)**(11/2)*(3*a**2*c*d*e**4 - 6*a*c**2*d**3*e**2 + 3*c**3*d**5)/(11*e**3) + (d + e*x)**(9/2)*(a**3*e**6 -
3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 - c**3*d**6)/(9*e**3))/e

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Giac [B]  time = 1.26956, size = 1258, normalized size = 10.57 \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]

2/45045*(143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*
c^3*d^6*e^(-3) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*c^2*d^5*e^(-1) +
39*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155
*(x*e + d)^(3/2)*d^4)*c^3*d^5*e^(-3) + 9009*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*c*d^4*e + 1287*(35*(
x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*c^2*d^4*e^(-1) +
15*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 +
9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*c^3*d^4*e^(-3) + 15015*(x*e + d)^(3/2)*a^3*d^3*e^3 + 3861
*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*c*d^3*e + 117*(315*(x*e + d)^(11/2)
- 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a*c
^2*d^3*e^(-1) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e +
d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*c^3*d^3*e^(
-3) + 9009*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*d^2*e^3 + 1287*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7
/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^2*c*d^2*e + 15*(693*(x*e + d)^(13/2) - 4095*(x*e
+ d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e +
d)^(3/2)*d^5)*a*c^2*d^2*e^(-1) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^
3*d*e^3 + 39*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*
d^3 + 1155*(x*e + d)^(3/2)*d^4)*a^2*c*d*e + 143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5
/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^3*e^3)*e^(-1)