3.20.94 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{3/2}} \, dx\) [1994]

Optimal. Leaf size=119 \[ -\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}{5 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{7/2}}{7 e^4}-\frac {2 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{9/2}}{3 e^4}+\frac {2 c^3 d^3 (d+e x)^{11/2}}{11 e^4} \]

[Out]

-2/5*(-a*e^2+c*d^2)^3*(e*x+d)^(5/2)/e^4+6/7*c*d*(-a*e^2+c*d^2)^2*(e*x+d)^(7/2)/e^4-2/3*c^2*d^2*(-a*e^2+c*d^2)*
(e*x+d)^(9/2)/e^4+2/11*c^3*d^3*(e*x+d)^(11/2)/e^4

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Rubi [A]
time = 0.04, 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 {2 c^2 d^2 (d+e x)^{9/2} \left (c d^2-a e^2\right )}{3 e^4}+\frac {6 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^4}-\frac {2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3}{5 e^4}+\frac {2 c^3 d^3 (d+e x)^{11/2}}{11 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.07, size = 111, normalized size = 0.93 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (231 a^3 e^6-99 a^2 c d e^4 (2 d-5 e x)+11 a c^2 d^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(231*a^3*e^6 - 99*a^2*c*d*e^4*(2*d - 5*e*x) + 11*a*c^2*d^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x
^2) + c^3*d^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)))/(1155*e^4)

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Maple [A]
time = 0.71, size = 97, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\)	\(97\)
default	\(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\)	\(97\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (105 c^{3} d^{3} e^{3} x^{3}+385 a \,c^{2} d^{2} e^{4} x^{2}-70 c^{3} d^{4} e^{2} x^{2}+495 a^{2} c d \,e^{5} x -220 a \,c^{2} d^{3} e^{3} x +40 c^{3} d^{5} e x +231 e^{6} a^{3}-198 e^{4} d^{2} a^{2} c +88 d^{4} e^{2} c^{2} a -16 d^{6} c^{3}\right )}{1155 e^{4}}\)	\(131\)
trager	\(\frac {2 \left (105 c^{3} d^{3} e^{5} x^{5}+385 c^{2} d^{2} a \,e^{6} x^{4}+140 c^{3} d^{4} e^{4} x^{4}+495 d \,e^{7} a^{2} c \,x^{3}+550 c^{2} d^{3} a \,e^{5} x^{3}+5 c^{3} d^{5} e^{3} x^{3}+231 e^{8} a^{3} x^{2}+792 d^{2} e^{6} a^{2} c \,x^{2}+33 c^{2} d^{4} a \,e^{4} x^{2}-6 c^{3} d^{6} e^{2} x^{2}+462 a^{3} d \,e^{7} x +99 a^{2} c \,d^{3} e^{5} x -44 a \,c^{2} d^{5} e^{3} x +8 c^{3} d^{7} e x +231 e^{6} a^{3} d^{2}-198 d^{4} e^{4} a^{2} c +88 c^{2} d^{6} a \,e^{2}-16 c^{3} d^{8}\right ) \sqrt {e x +d}}{1155 e^{4}}\)	\(243\)
risch	\(\frac {2 \left (105 c^{3} d^{3} e^{5} x^{5}+385 c^{2} d^{2} a \,e^{6} x^{4}+140 c^{3} d^{4} e^{4} x^{4}+495 d \,e^{7} a^{2} c \,x^{3}+550 c^{2} d^{3} a \,e^{5} x^{3}+5 c^{3} d^{5} e^{3} x^{3}+231 e^{8} a^{3} x^{2}+792 d^{2} e^{6} a^{2} c \,x^{2}+33 c^{2} d^{4} a \,e^{4} x^{2}-6 c^{3} d^{6} e^{2} x^{2}+462 a^{3} d \,e^{7} x +99 a^{2} c \,d^{3} e^{5} x -44 a \,c^{2} d^{5} e^{3} x +8 c^{3} d^{7} e x +231 e^{6} a^{3} d^{2}-198 d^{4} e^{4} a^{2} c +88 c^{2} d^{6} a \,e^{2}-16 c^{3} d^{8}\right ) \sqrt {e x +d}}{1155 e^{4}}\)	\(243\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/e^4*(1/11*c^3*d^3*(e*x+d)^(11/2)+1/3*(a*e^2-c*d^2)*c^2*d^2*(e*x+d)^(9/2)+3/7*(a*e^2-c*d^2)^2*c*d*(e*x+d)^(7/
2)+1/5*(a*e^2-c*d^2)^3*(e*x+d)^(5/2))

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Maxima [A]
time = 0.29, size = 134, normalized size = 1.13 \begin {gather*} \frac {2}{1155} \, {\left (105 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{3} d^{3} - 385 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 495 \, {\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 {7}{2}} - 231 \, {\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 {5}{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)^(3/2),x, algorithm="maxima")

[Out]

2/1155*(105*(x*e + d)^(11/2)*c^3*d^3 - 385*(c^3*d^4 - a*c^2*d^2*e^2)*(x*e + d)^(9/2) + 495*(c^3*d^5 - 2*a*c^2*
d^3*e^2 + a^2*c*d*e^4)*(x*e + d)^(7/2) - 231*(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)
^(5/2))*e^(-4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (100) = 200\).
time = 2.44, size = 224, normalized size = 1.88 \begin {gather*} \frac {2}{1155} \, {\left (8 \, c^{3} d^{7} x e - 16 \, c^{3} d^{8} + 231 \, a^{3} x^{2} e^{8} + 33 \, {\left (15 \, a^{2} c d x^{3} + 14 \, a^{3} d x\right )} e^{7} + 11 \, {\left (35 \, a c^{2} d^{2} x^{4} + 72 \, a^{2} c d^{2} x^{2} + 21 \, a^{3} d^{2}\right )} e^{6} + {\left (105 \, c^{3} d^{3} x^{5} + 550 \, a c^{2} d^{3} x^{3} + 99 \, a^{2} c d^{3} x\right )} e^{5} + {\left (140 \, c^{3} d^{4} x^{4} + 33 \, a c^{2} d^{4} x^{2} - 198 \, a^{2} c d^{4}\right )} e^{4} + {\left (5 \, c^{3} d^{5} x^{3} - 44 \, a c^{2} d^{5} x\right )} e^{3} - 2 \, {\left (3 \, c^{3} d^{6} x^{2} - 44 \, a c^{2} d^{6}\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)^(3/2),x, algorithm="fricas")

[Out]

2/1155*(8*c^3*d^7*x*e - 16*c^3*d^8 + 231*a^3*x^2*e^8 + 33*(15*a^2*c*d*x^3 + 14*a^3*d*x)*e^7 + 11*(35*a*c^2*d^2
*x^4 + 72*a^2*c*d^2*x^2 + 21*a^3*d^2)*e^6 + (105*c^3*d^3*x^5 + 550*a*c^2*d^3*x^3 + 99*a^2*c*d^3*x)*e^5 + (140*
c^3*d^4*x^4 + 33*a*c^2*d^4*x^2 - 198*a^2*c*d^4)*e^4 + (5*c^3*d^5*x^3 - 44*a*c^2*d^5*x)*e^3 - 2*(3*c^3*d^6*x^2
- 44*a*c^2*d^6)*e^2)*sqrt(x*e + d)*e^(-4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 971 vs. \(2 (110) = 220\).
time = 47.94, size = 971, normalized size = 8.16 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{3} d^{3} e^{3}}{\sqrt {d + e x}} - 6 a^{3} d^{2} e^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 6 a^{3} d e^{3} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right ) - 2 a^{3} e^{3} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right ) - 6 a^{2} c d^{4} e \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 18 a^{2} c d^{3} e \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right ) - 18 a^{2} c d^{2} e \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right ) - 6 a^{2} c d e \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right ) - \frac {6 a c^{2} d^{5} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {18 a c^{2} d^{4} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} - \frac {18 a c^{2} d^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e} - \frac {6 a c^{2} d^{2} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e} - \frac {2 c^{3} d^{6} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {6 c^{3} d^{5} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {6 c^{3} d^{4} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} - \frac {2 c^{3} d^{3} \left (\frac {d^{6}}{\sqrt {d + e x}} + 6 d^{5} \sqrt {d + e x} - 5 d^{4} \left (d + e x\right )^{\frac {3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac {5}{2}} - \frac {15 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {2 d \left (d + e x\right )^{\frac {9}{2}}}{3} - \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{3}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {9}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \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)**(3/2),x)

[Out]

Piecewise(((-2*a**3*d**3*e**3/sqrt(d + e*x) - 6*a**3*d**2*e**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 6*a**3*d*e
**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) - 2*a**3*e**3*(-d**3/sqrt(d + e*x) - 3*d**2*
sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5) - 6*a**2*c*d**4*e*(-d/sqrt(d + e*x) - sqrt(d + e*x))
- 18*a**2*c*d**3*e*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) - 18*a**2*c*d**2*e*(-d**3/sqr
t(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5) - 6*a**2*c*d*e*(d**4/sqrt(d + e*x
) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7) - 6*a*c**2*d
**5*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 18*a*c**2*d**4*(-d**3/sqrt(d + e*x) - 3*
d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e - 18*a*c**2*d**3*(d**4/sqrt(d + e*x) + 4*d**3*
sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e - 6*a*c**2*d**2*(-d**
5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)*
*(7/2)/7 - (d + e*x)**(9/2)/9)/e - 2*c**3*d**6*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2
) - (d + e*x)**(5/2)/5)/e**3 - 6*c**3*d**5*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2
) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 6*c**3*d**4*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x
) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**3 -
2*c**3*d**3*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) -
15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**3)/e, Ne(e, 0)), (c**3*d**(9/2)
*x**4/4, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (100) = 200\).
time = 1.18, size = 617, normalized size = 5.18 \begin {gather*} \frac {2}{3465} \, {\left (99 \, {\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^{5} e^{\left (-3\right )} + 693 \, {\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^{4} e^{\left (-1\right )} + 22 \, {\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^{4} e^{\left (-3\right )} + 3465 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} c d^{3} e + 594 \, {\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^{3} e^{\left (-1\right )} + 5 \, {\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^{3} e^{\left (-3\right )} + 1386 \, {\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^{2} e + 33 \, {\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^{2} e^{\left (-1\right )} + 3465 \, \sqrt {x e + d} a^{3} d^{2} e^{3} + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} d e^{3} + 297 \, {\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 e + 231 \, {\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} 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)^(3/2),x, algorithm="giac")

[Out]

2/3465*(99*(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^5*
e^(-3) + 693*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c^2*d^4*e^(-1) + 22*(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^4*e^(-3) + 3465*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*c*d^3*e + 594*(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^3*e^(-1) + 5*(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^3*e^(-3) + 1386*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*d^2*
e + 33*(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*s
qrt(x*e + d)*d^4)*a*c^2*d^2*e^(-1) + 3465*sqrt(x*e + d)*a^3*d^2*e^3 + 2310*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*
d)*a^3*d*e^3 + 297*(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*e + 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*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 )}^{5/2}}{5\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a*e^2 - c*d^2)^3*(d + e*x)^(5/2))/(5*e^4) - ((6*c^3*d^4 - 6*a*c^2*d^2*e^2)*(d + e*x)^(9/2))/(9*e^4) + (2*c
^3*d^3*(d + e*x)^(11/2))/(11*e^4) + (6*c*d*(a*e^2 - c*d^2)^2*(d + e*x)^(7/2))/(7*e^4)

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