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

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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 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])

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]

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 = 0.09, size = 111, normalized size = 0.93 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (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 \left (8 d^2-36 d e x+99 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )}{6435 e^4} \end {gather*}

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)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.12, size = 151, normalized size = 1.27 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (715 a^3 e^6-2145 a^2 c d^2 e^4+1755 a^2 c d e^4 (d+e x)+2145 a c^2 d^4 e^2-3510 a c^2 d^3 e^2 (d+e x)+1485 a c^2 d^2 e^2 (d+e x)^2-715 c^3 d^6+1755 c^3 d^5 (d+e x)-1485 c^3 d^4 (d+e x)^2+429 c^3 d^3 (d+e x)^3\right )}{6435 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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*c^3*d^6 + 2145*a*c^2*d^4*e^2 - 2145*a^2*c*d^2*e^4 + 715*a^3*e^6 + 1755*c^3*d^5*(d + e
*x) - 3510*a*c^2*d^3*e^2*(d + e*x) + 1755*a^2*c*d*e^4*(d + e*x) - 1485*c^3*d^4*(d + e*x)^2 + 1485*a*c^2*d^2*e^
2*(d + e*x)^2 + 429*c^3*d^3*(d + e*x)^3))/(6435*e^4)

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 335, normalized size = 2.82 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

giac [B]  time = 0.29, size = 1293, normalized size = 10.87

result too large to display

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*(1287*(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
^7*e^(-3) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c^2*d^6*e^(-1) + 572*(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^6*e^(-3) + 45045*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*c*d^5*e + 15444*(5*(x*e + d)^(7/2) - 2
1*(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^5*e^(-1) + 390*(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 - 6
93*sqrt(x*e + d)*d^5)*c^3*d^5*e^(-3) + 36036*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)
*a^2*c*d^4*e + 2574*(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^4*e^(-1) + 60*(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^4*e^(-3) + 45045*sqrt(x*e + d)*a^3*d^4*e^3 + 60060*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*
a^3*d^3*e^3 + 23166*(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^3*e + 780*(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^3*e^(-1) + 7*(429*(x*e + d)^(15/2) - 3465*(
x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027
*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^3*d^3*e^(-3) + 18018*(3*(x*e + d)
^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*d^2*e^3 + 1716*(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^2*e + 45*(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)*a*c^2*d^2*e^(-1) + 5148*(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*d*e^3 + 195*(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^2*c*d*e + 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^3*e^3)*e^(-1)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 131, normalized size = 1.10 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (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}\right )}{6435 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^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

________________________________________________________________________________________

maxima [A]  time = 1.01, size = 137, normalized size = 1.15 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 106, normalized size = 0.89 \begin {gather*} \frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4} \end {gather*}

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,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 5.41, size = 165, normalized size = 1.39 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________