3.21.37 \(\int (d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\) [2037]

Optimal. Leaf size=233 \[ \frac {32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 c^4 d^4 (d+e x)^{5/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d} \]

[Out]

32/1155*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^4/d^4/(e*x+d)^(5/2)+16/231*(-a*e^2+c*d^2)^2
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3/(e*x+d)^(3/2)+4/33*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(5/2)/c^2/d^2/(e*x+d)^(1/2)+2/11*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*(e*x+d)^(1/2)/c/d

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Rubi [A]
time = 0.12, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662} \begin {gather*} \frac {32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 c^4 d^4 (d+e x)^{5/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(32*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(1155*c^4*d^4*(d + e*x)^(5/2)) + (16*(c*d
^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(231*c^3*d^3*(d + e*x)^(3/2)) + (4*(c*d^2 - a*e^2
)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(33*c^2*d^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(5/2))/(11*c*d)

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

\begin {align*} \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx &=\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d}+\frac {\left (6 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{11 d}\\ &=\frac {4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{33 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{231 d^3}\\ &=\frac {32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 c^4 d^4 (d+e x)^{5/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 132, normalized size = 0.57 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2} \left (-16 a^3 e^6+8 a^2 c d e^4 (11 d+5 e x)-2 a c^2 d^2 e^2 \left (99 d^2+110 d e x+35 e^2 x^2\right )+c^3 d^3 \left (231 d^3+495 d^2 e x+385 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 c^4 d^4 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(11*d + 5*e*x) - 2*a*c^2*d^2*e^2*(99*d^2 + 110
*d*e*x + 35*e^2*x^2) + c^3*d^3*(231*d^3 + 495*d^2*e*x + 385*d*e^2*x^2 + 105*e^3*x^3)))/(1155*c^4*d^4*(d + e*x)
^(5/2))

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Maple [A]
time = 0.84, size = 160, normalized size = 0.69

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{2} \left (-105 c^{3} d^{3} e^{3} x^{3}+70 a \,c^{2} d^{2} e^{4} x^{2}-385 c^{3} d^{4} e^{2} x^{2}-40 a^{2} c d \,e^{5} x +220 a \,c^{2} d^{3} e^{3} x -495 c^{3} d^{5} e x +16 e^{6} a^{3}-88 e^{4} d^{2} a^{2} c +198 d^{4} e^{2} c^{2} a -231 d^{6} c^{3}\right )}{1155 \sqrt {e x +d}\, c^{4} d^{4}}\)	\(160\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-105 c^{3} d^{3} e^{3} x^{3}+70 a \,c^{2} d^{2} e^{4} x^{2}-385 c^{3} d^{4} e^{2} x^{2}-40 a^{2} c d \,e^{5} x +220 a \,c^{2} d^{3} e^{3} x -495 c^{3} d^{5} e x +16 e^{6} a^{3}-88 e^{4} d^{2} a^{2} c +198 d^{4} e^{2} c^{2} a -231 d^{6} c^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 c^{4} d^{4} \left (e x +d \right )^{\frac {3}{2}}}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1155/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(c*d*x+a*e)^2*(-105*c^3*d^3*e^3*x^3+70*a*c^2*d^2*e^4*x^2-385
*c^3*d^4*e^2*x^2-40*a^2*c*d*e^5*x+220*a*c^2*d^3*e^3*x-495*c^3*d^5*e*x+16*a^3*e^6-88*a^2*c*d^2*e^4+198*a*c^2*d^
4*e^2-231*c^3*d^6)/c^4/d^4

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Maxima [A]
time = 0.30, size = 263, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left (105 \, c^{5} d^{5} x^{5} e^{3} + 231 \, a^{2} c^{3} d^{6} e^{2} - 198 \, a^{3} c^{2} d^{4} e^{4} + 88 \, a^{4} c d^{2} e^{6} - 16 \, a^{5} e^{8} + 35 \, {\left (11 \, c^{5} d^{6} e^{2} + 4 \, a c^{4} d^{4} e^{4}\right )} x^{4} + 5 \, {\left (99 \, c^{5} d^{7} e + 110 \, a c^{4} d^{5} e^{3} + a^{2} c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (77 \, c^{5} d^{8} + 264 \, a c^{4} d^{6} e^{2} + 11 \, a^{2} c^{3} d^{4} e^{4} - 2 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + {\left (462 \, a c^{4} d^{7} e + 99 \, a^{2} c^{3} d^{5} e^{3} - 44 \, a^{3} c^{2} d^{3} e^{5} + 8 \, a^{4} c d e^{7}\right )} x\right )} \sqrt {c d x + a e} {\left (x e + d\right )}}{1155 \, {\left (c^{4} d^{4} x e + c^{4} d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*c^5*d^5*x^5*e^3 + 231*a^2*c^3*d^6*e^2 - 198*a^3*c^2*d^4*e^4 + 88*a^4*c*d^2*e^6 - 16*a^5*e^8 + 35*(
11*c^5*d^6*e^2 + 4*a*c^4*d^4*e^4)*x^4 + 5*(99*c^5*d^7*e + 110*a*c^4*d^5*e^3 + a^2*c^3*d^3*e^5)*x^3 + 3*(77*c^5
*d^8 + 264*a*c^4*d^6*e^2 + 11*a^2*c^3*d^4*e^4 - 2*a^3*c^2*d^2*e^6)*x^2 + (462*a*c^4*d^7*e + 99*a^2*c^3*d^5*e^3
- 44*a^3*c^2*d^3*e^5 + 8*a^4*c*d*e^7)*x)*sqrt(c*d*x + a*e)*(x*e + d)/(c^4*d^4*x*e + c^4*d^5)

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Fricas [A]
time = 2.79, size = 289, normalized size = 1.24 \begin {gather*} \frac {2 \, {\left (231 \, c^{5} d^{8} x^{2} + 8 \, a^{4} c d x e^{7} - 16 \, a^{5} e^{8} - 2 \, {\left (3 \, a^{3} c^{2} d^{2} x^{2} - 44 \, a^{4} c d^{2}\right )} e^{6} + {\left (5 \, a^{2} c^{3} d^{3} x^{3} - 44 \, a^{3} c^{2} d^{3} x\right )} e^{5} + {\left (140 \, a c^{4} d^{4} x^{4} + 33 \, a^{2} c^{3} d^{4} x^{2} - 198 \, a^{3} c^{2} d^{4}\right )} e^{4} + {\left (105 \, c^{5} d^{5} x^{5} + 550 \, a c^{4} d^{5} x^{3} + 99 \, a^{2} c^{3} d^{5} x\right )} e^{3} + 11 \, {\left (35 \, c^{5} d^{6} x^{4} + 72 \, a c^{4} d^{6} x^{2} + 21 \, a^{2} c^{3} d^{6}\right )} e^{2} + 33 \, {\left (15 \, c^{5} d^{7} x^{3} + 14 \, a c^{4} d^{7} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{1155 \, {\left (c^{4} d^{4} x e + c^{4} d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1728 vs. \(2 (214) = 428\).
time = 1.23, size = 1728, normalized size = 7.42 \begin {gather*} -\frac {2}{3465} \, {\left (231 \, c d^{4} {\left (\frac {{\left (5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} e^{\left (-2\right )}}{c^{2} d^{2}} + \frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} - \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} - 2 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}}{c^{2} d^{2}}\right )} e^{\left (-2\right )} - 1155 \, a d^{3} {\left (\frac {{\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{\left (-1\right )}}{c d} + \frac {\sqrt {-c d^{2} e + a e^{3}} c d^{2} - \sqrt {-c d^{2} e + a e^{3}} a e^{2}}{c d}\right )} - 99 \, c d^{3} {\left (\frac {{\left (15 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6}\right )} e^{\left (-2\right )}}{c^{3} d^{3}} + \frac {{\left (35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} e^{\left (-5\right )}}{c^{3} d^{3}}\right )} + 33 \, c d^{2} {\left (\frac {{\left (35 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{8} - 5 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{6} e^{2} - 6 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{2} d^{4} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c d^{2} e^{6} - 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} e^{8}\right )} e^{\left (-3\right )}}{c^{4} d^{4}} + \frac {{\left (105 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} e^{9} - 189 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} e^{6} + 135 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a e^{3} - 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}}\right )} e^{\left (-7\right )}}{c^{4} d^{4}}\right )} e + 693 \, a d^{2} {\left (\frac {{\left (5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} e^{\left (-2\right )}}{c^{2} d^{2}} + \frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} - \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} - 2 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}}{c^{2} d^{2}}\right )} - 99 \, a d {\left (\frac {{\left (15 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6}\right )} e^{\left (-2\right )}}{c^{3} d^{3}} + \frac {{\left (35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} e^{\left (-5\right )}}{c^{3} d^{3}}\right )} e^{2} - c d {\left (\frac {{\left (315 \, \sqrt {-c d^{2} e + a e^{3}} c^{5} d^{10} - 35 \, \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{8} e^{2} - 40 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{3} d^{6} e^{4} - 48 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c^{2} d^{4} e^{6} - 64 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} c d^{2} e^{8} - 128 \, \sqrt {-c d^{2} e + a e^{3}} a^{5} e^{10}\right )} e^{\left (-4\right )}}{c^{5} d^{5}} + \frac {{\left (1155 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{4} e^{12} - 2772 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{3} e^{9} + 2970 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a^{2} e^{6} - 1540 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} a e^{3} + 315 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {11}{2}}\right )} e^{\left (-9\right )}}{c^{5} d^{5}}\right )} e^{2} + 11 \, a {\left (\frac {{\left (35 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{8} - 5 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{6} e^{2} - 6 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{2} d^{4} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c d^{2} e^{6} - 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} e^{8}\right )} e^{\left (-3\right )}}{c^{4} d^{4}} + \frac {{\left (105 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} e^{9} - 189 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} e^{6} + 135 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a e^{3} - 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}}\right )} e^{\left (-7\right )}}{c^{4} d^{4}}\right )} e^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(231*c*d^4*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)
^(5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*
d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2))*e^(-2) - 1155*a*d^3*(((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*e^(-1)/(c*d)
+ (sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^2*e + a*e^3)*a*e^2)/(c*d)) - 99*c*d^3*((15*sqrt(-c*d^2*e + a*e^3)
*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e +
a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*
e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3)) + 33*c*d^2*
((35*sqrt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^
2*d^4*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (10
5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 +
135*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^
4*d^4))*e + 693*a*d^2*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*e + d)*c*d*e - c*d^2*e + a*e
^3)^(5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(
-c*d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2)) - 99*a*d*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*
a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) +
(35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 1
5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^2 - c*d*((315*sqrt(-c*d^2*e + a*e^3)*c^5*d^10
- 35*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2 - 40*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^6*e^4 - 48*sqrt(-c*d^2*e + a*
e^3)*a^3*c^2*d^4*e^6 - 64*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8 - 128*sqrt(-c*d^2*e + a*e^3)*a^5*e^10)*e^(-4)/(
c^5*d^5) + (1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((x*e + d)*c*d*e - c*d^2*e + a*e^3)
^(5/2)*a^3*e^9 + 2970*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((x*e + d)*c*d*e - c*d^2*e + a*
e^3)^(9/2)*a*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))*e^(-9)/(c^5*d^5))*e^2 + 11*a*((35*sqrt(-c*d
^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*
sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c
*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)
*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4))*e^3)*e
^(-1)

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Mupad [B]
time = 1.21, size = 320, normalized size = 1.37 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e\,x^4\,\left (11\,c\,d^2+4\,a\,e^2\right )\,\sqrt {d+e\,x}}{33}+\frac {2\,c\,d\,e^2\,x^5\,\sqrt {d+e\,x}}{11}-\frac {\sqrt {d+e\,x}\,\left (32\,a^5\,e^8-176\,a^4\,c\,d^2\,e^6+396\,a^3\,c^2\,d^4\,e^4-462\,a^2\,c^3\,d^6\,e^2\right )}{1155\,c^4\,d^4\,e}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (a^2\,e^4+110\,a\,c\,d^2\,e^2+99\,c^2\,d^4\right )}{231\,c\,d}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-12\,a^3\,c^2\,d^2\,e^6+66\,a^2\,c^3\,d^4\,e^4+1584\,a\,c^4\,d^6\,e^2+462\,c^5\,d^8\right )}{1155\,c^4\,d^4\,e}+\frac {2\,a\,x\,\sqrt {d+e\,x}\,\left (8\,a^3\,e^6-44\,a^2\,c\,d^2\,e^4+99\,a\,c^2\,d^4\,e^2+462\,c^3\,d^6\right )}{1155\,c^3\,d^3}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*e*x^4*(4*a*e^2 + 11*c*d^2)*(d + e*x)^(1/2))/33 + (2*c*d*e^2
*x^5*(d + e*x)^(1/2))/11 - ((d + e*x)^(1/2)*(32*a^5*e^8 - 176*a^4*c*d^2*e^6 - 462*a^2*c^3*d^6*e^2 + 396*a^3*c^
2*d^4*e^4))/(1155*c^4*d^4*e) + (2*x^3*(d + e*x)^(1/2)*(a^2*e^4 + 99*c^2*d^4 + 110*a*c*d^2*e^2))/(231*c*d) + (x
^2*(d + e*x)^(1/2)*(462*c^5*d^8 + 1584*a*c^4*d^6*e^2 + 66*a^2*c^3*d^4*e^4 - 12*a^3*c^2*d^2*e^6))/(1155*c^4*d^4
*e) + (2*a*x*(d + e*x)^(1/2)*(8*a^3*e^6 + 462*c^3*d^6 + 99*a*c^2*d^4*e^2 - 44*a^2*c*d^2*e^4))/(1155*c^3*d^3)))
/(x + d/e)

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