3.20.92 \(\int \sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx\) [1992]

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

[Out]

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

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Rubi [A]
time = 0.06, 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 = {640, 45} \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 45

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 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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*}

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Mathematica [A]
time = 0.08, 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)

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Maple [A]
time = 0.69, size = 97, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\)	\(97\)
default	\(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\)	\(97\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (429 c^{3} d^{3} e^{3} x^{3}+1485 a \,c^{2} d^{2} e^{4} x^{2}-198 c^{3} d^{4} e^{2} x^{2}+1755 a^{2} c d \,e^{5} x -540 a \,c^{2} d^{3} e^{3} x +72 c^{3} d^{5} e x +715 e^{6} a^{3}-390 e^{4} d^{2} a^{2} c +120 d^{4} e^{2} c^{2} a -16 d^{6} c^{3}\right )}{6435 e^{4}}\)	\(131\)
trager	\(\frac {2 \left (429 c^{3} d^{3} e^{7} x^{7}+1485 a \,c^{2} d^{2} e^{8} x^{6}+1518 c^{3} d^{4} e^{6} x^{6}+1755 a^{2} c d \,e^{9} x^{5}+5400 a \,c^{2} d^{3} e^{7} x^{5}+1854 c^{3} d^{5} e^{5} x^{5}+715 a^{3} e^{10} x^{4}+6630 a^{2} c \,d^{2} e^{8} x^{4}+6870 a \,c^{2} d^{4} e^{6} x^{4}+800 c^{3} d^{6} e^{4} x^{4}+2860 a^{3} d \,e^{9} x^{3}+8970 a^{2} c \,d^{3} e^{7} x^{3}+3180 a \,c^{2} d^{5} e^{5} x^{3}+5 c^{3} d^{7} e^{3} x^{3}+4290 a^{3} d^{2} e^{8} x^{2}+4680 a^{2} c \,d^{4} e^{6} x^{2}+45 a \,c^{2} d^{6} e^{4} x^{2}-6 c^{3} d^{8} e^{2} x^{2}+2860 a^{3} d^{3} e^{7} x +195 a^{2} c \,d^{5} e^{5} x -60 a \,c^{2} d^{7} e^{3} x +8 c^{3} d^{9} e x +715 a^{3} d^{4} e^{6}-390 a^{2} c \,d^{6} e^{4}+120 a \,c^{2} d^{8} e^{2}-16 c^{3} d^{10}\right ) \sqrt {e x +d}}{6435 e^{4}}\)	\(359\)
risch	\(\frac {2 \left (429 c^{3} d^{3} e^{7} x^{7}+1485 a \,c^{2} d^{2} e^{8} x^{6}+1518 c^{3} d^{4} e^{6} x^{6}+1755 a^{2} c d \,e^{9} x^{5}+5400 a \,c^{2} d^{3} e^{7} x^{5}+1854 c^{3} d^{5} e^{5} x^{5}+715 a^{3} e^{10} x^{4}+6630 a^{2} c \,d^{2} e^{8} x^{4}+6870 a \,c^{2} d^{4} e^{6} x^{4}+800 c^{3} d^{6} e^{4} x^{4}+2860 a^{3} d \,e^{9} x^{3}+8970 a^{2} c \,d^{3} e^{7} x^{3}+3180 a \,c^{2} d^{5} e^{5} x^{3}+5 c^{3} d^{7} e^{3} x^{3}+4290 a^{3} d^{2} e^{8} x^{2}+4680 a^{2} c \,d^{4} e^{6} x^{2}+45 a \,c^{2} d^{6} e^{4} x^{2}-6 c^{3} d^{8} e^{2} x^{2}+2860 a^{3} d^{3} e^{7} x +195 a^{2} c \,d^{5} e^{5} x -60 a \,c^{2} d^{7} e^{3} x +8 c^{3} d^{9} e x +715 a^{3} d^{4} e^{6}-390 a^{2} c \,d^{6} e^{4}+120 a \,c^{2} d^{8} e^{2}-16 c^{3} d^{10}\right ) \sqrt {e x +d}}{6435 e^{4}}\)	\(359\)

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,method=_RETURNVERBOSE)

[Out]

2/e^4*(1/15*c^3*d^3*(e*x+d)^(15/2)+3/13*(a*e^2-c*d^2)*c^2*d^2*(e*x+d)^(13/2)+3/11*(a*e^2-c*d^2)^2*c*d*(e*x+d)^
(11/2)+1/9*(a*e^2-c*d^2)^3*(e*x+d)^(9/2))

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Maxima [A]
time = 0.30, size = 134, normalized size = 1.13 \begin {gather*} \frac {2}{6435} \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} c^{3} d^{3} - 1485 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (x e + 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 (x e + 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 (x e + d\right )}^{\frac {9}{2}}\right )} e^{\left (-4\right )} \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*(x*e + d)^(15/2)*c^3*d^3 - 1485*(c^3*d^4 - a*c^2*d^2*e^2)*(x*e + d)^(13/2) + 1755*(c^3*d^5 - 2*a*c
^2*d^3*e^2 + a^2*c*d*e^4)*(x*e + 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)*(x*e
+ d)^(9/2))*e^(-4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (100) = 200\).
time = 3.47, size = 325, normalized size = 2.73 \begin {gather*} \frac {2}{6435} \, {\left (8 \, c^{3} d^{9} x e - 16 \, c^{3} d^{10} + 715 \, a^{3} x^{4} e^{10} + 65 \, {\left (27 \, a^{2} c d x^{5} + 44 \, a^{3} d x^{3}\right )} e^{9} + 15 \, {\left (99 \, a c^{2} d^{2} x^{6} + 442 \, a^{2} c d^{2} x^{4} + 286 \, a^{3} d^{2} x^{2}\right )} e^{8} + {\left (429 \, c^{3} d^{3} x^{7} + 5400 \, a c^{2} d^{3} x^{5} + 8970 \, a^{2} c d^{3} x^{3} + 2860 \, a^{3} d^{3} x\right )} e^{7} + {\left (1518 \, c^{3} d^{4} x^{6} + 6870 \, a c^{2} d^{4} x^{4} + 4680 \, a^{2} c d^{4} x^{2} + 715 \, a^{3} d^{4}\right )} e^{6} + 3 \, {\left (618 \, c^{3} d^{5} x^{5} + 1060 \, a c^{2} d^{5} x^{3} + 65 \, a^{2} c d^{5} x\right )} e^{5} + 5 \, {\left (160 \, c^{3} d^{6} x^{4} + 9 \, a c^{2} d^{6} x^{2} - 78 \, a^{2} c d^{6}\right )} e^{4} + 5 \, {\left (c^{3} d^{7} x^{3} - 12 \, a c^{2} d^{7} x\right )} e^{3} - 6 \, {\left (c^{3} d^{8} x^{2} - 20 \, a c^{2} d^{8}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-4\right )} \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*(8*c^3*d^9*x*e - 16*c^3*d^10 + 715*a^3*x^4*e^10 + 65*(27*a^2*c*d*x^5 + 44*a^3*d*x^3)*e^9 + 15*(99*a*c^2
*d^2*x^6 + 442*a^2*c*d^2*x^4 + 286*a^3*d^2*x^2)*e^8 + (429*c^3*d^3*x^7 + 5400*a*c^2*d^3*x^5 + 8970*a^2*c*d^3*x
^3 + 2860*a^3*d^3*x)*e^7 + (1518*c^3*d^4*x^6 + 6870*a*c^2*d^4*x^4 + 4680*a^2*c*d^4*x^2 + 715*a^3*d^4)*e^6 + 3*
(618*c^3*d^5*x^5 + 1060*a*c^2*d^5*x^3 + 65*a^2*c*d^5*x)*e^5 + 5*(160*c^3*d^6*x^4 + 9*a*c^2*d^6*x^2 - 78*a^2*c*
d^6)*e^4 + 5*(c^3*d^7*x^3 - 12*a*c^2*d^7*x)*e^3 - 6*(c^3*d^8*x^2 - 20*a*c^2*d^8)*e^2)*sqrt(x*e + d)*e^(-4)

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Sympy [A]
time = 2.01, 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}} \cdot \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}} \cdot \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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1293 vs. \(2 (100) = 200\).
time = 1.76, size = 1293, normalized size = 10.87 \begin {gather*} \frac {2}{45045} \, {\left (1287 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c^{3} d^{7} e^{\left (-3\right )} + 9009 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a c^{2} d^{6} e^{\left (-1\right )} + 572 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c^{3} d^{6} e^{\left (-3\right )} + 45045 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} c d^{5} e + 15444 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a c^{2} d^{5} e^{\left (-1\right )} + 390 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c^{3} d^{5} e^{\left (-3\right )} + 36036 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} c d^{4} e + 2574 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a c^{2} d^{4} e^{\left (-1\right )} + 60 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{3} d^{4} e^{\left (-3\right )} + 45045 \, \sqrt {x e + d} a^{3} d^{4} e^{3} + 60060 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} d^{3} e^{3} + 23166 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} c d^{3} e + 780 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a c^{2} d^{3} e^{\left (-1\right )} + 7 \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} c^{3} d^{3} e^{\left (-3\right )} + 18018 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{3} d^{2} e^{3} + 1716 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{2} c d^{2} e + 45 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} a c^{2} d^{2} e^{\left (-1\right )} + 5148 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{3} d e^{3} + 195 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a^{2} c d e + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{3} e^{3}\right )} e^{\left (-1\right )} \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="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)

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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)

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