3.20.98 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{11/2}} \, dx\) [1998]

Optimal. Leaf size=115 \[ \frac {2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}-\frac {6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt {d+e x}}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{e^4}+\frac {2 c^3 d^3 (d+e x)^{3/2}}{3 e^4} \]

[Out]

2/3*(-a*e^2+c*d^2)^3/e^4/(e*x+d)^(3/2)+2/3*c^3*d^3*(e*x+d)^(3/2)/e^4-6*c*d*(-a*e^2+c*d^2)^2/e^4/(e*x+d)^(1/2)-
6*c^2*d^2*(-a*e^2+c*d^2)*(e*x+d)^(1/2)/e^4

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Rubi [A]
time = 0.04, antiderivative size = 115, 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 \sqrt {d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac {6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt {d+e x}}+\frac {2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac {2 c^3 d^3 (d+e x)^{3/2}}{3 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)^(11/2),x]

[Out]

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

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Mathematica [A]
time = 0.08, size = 110, normalized size = 0.96 \begin {gather*} -\frac {2 \left (a^3 e^6+3 a^2 c d e^4 (2 d+3 e x)-3 a c^2 d^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^3 d^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \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)^(11/2),x]

[Out]

(-2*(a^3*e^6 + 3*a^2*c*d*e^4*(2*d + 3*e*x) - 3*a*c^2*d^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c^3*d^3*(16*d^3
+ 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)))/(3*e^4*(d + e*x)^(3/2))

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Maple [A]
time = 0.72, size = 141, normalized size = 1.23

method	result	size
risch	\(\frac {2 c^{2} d^{2} \left (c d e x +9 e^{2} a -8 c \,d^{2}\right ) \sqrt {e x +d}}{3 e^{4}}-\frac {2 \left (9 c d e x +e^{2} a +8 c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 e^{4} \left (e x +d \right )^{\frac {3}{2}}}\)	\(93\)
gosper	\(-\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-9 a \,c^{2} d^{2} e^{4} x^{2}+6 c^{3} d^{4} e^{2} x^{2}+9 a^{2} c d \,e^{5} x -36 a \,c^{2} d^{3} e^{3} x +24 c^{3} d^{5} e x +e^{6} a^{3}+6 e^{4} d^{2} a^{2} c -24 d^{4} e^{2} c^{2} a +16 d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\)	\(130\)
trager	\(-\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-9 a \,c^{2} d^{2} e^{4} x^{2}+6 c^{3} d^{4} e^{2} x^{2}+9 a^{2} c d \,e^{5} x -36 a \,c^{2} d^{3} e^{3} x +24 c^{3} d^{5} e x +e^{6} a^{3}+6 e^{4} d^{2} a^{2} c -24 d^{4} e^{2} c^{2} a +16 d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\)	\(130\)
derivativedivides	\(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}-6 c^{3} d^{4} \sqrt {e x +d}-\frac {6 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\)	\(141\)
default	\(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}-6 c^{3} d^{4} \sqrt {e x +d}-\frac {6 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\)	\(141\)

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)^(11/2),x,method=_RETURNVERBOSE)

[Out]

2/e^4*(1/3*c^3*d^3*(e*x+d)^(3/2)+3*a*c^2*d^2*e^2*(e*x+d)^(1/2)-3*c^3*d^4*(e*x+d)^(1/2)-3*c*d*(a^2*e^4-2*a*c*d^
2*e^2+c^2*d^4)/(e*x+d)^(1/2)-1/3*(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)/(e*x+d)^(3/2))

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Maxima [A]
time = 0.28, size = 136, normalized size = 1.18 \begin {gather*} \frac {2}{3} \, {\left ({\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} - 9 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \sqrt {x e + d}\right )} e^{\left (-3\right )} + \frac {{\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} - 9 \, {\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 )}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\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)^(11/2),x, algorithm="maxima")

[Out]

2/3*(((x*e + d)^(3/2)*c^3*d^3 - 9*(c^3*d^4 - a*c^2*d^2*e^2)*sqrt(x*e + d))*e^(-3) + (c^3*d^6 - 3*a*c^2*d^4*e^2
+ 3*a^2*c*d^2*e^4 - a^3*e^6 - 9*(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(x*e + d))*e^(-3)/(x*e + d)^(3/2))*
e^(-1)

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Fricas [A]
time = 2.81, size = 142, normalized size = 1.23 \begin {gather*} -\frac {2 \, {\left (24 \, c^{3} d^{5} x e + 16 \, c^{3} d^{6} + 9 \, a^{2} c d x e^{5} + a^{3} e^{6} - 3 \, {\left (3 \, a c^{2} d^{2} x^{2} - 2 \, a^{2} c d^{2}\right )} e^{4} - {\left (c^{3} d^{3} x^{3} + 36 \, a c^{2} d^{3} x\right )} e^{3} + 6 \, {\left (c^{3} d^{4} x^{2} - 4 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{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)^(11/2),x, algorithm="fricas")

[Out]

-2/3*(24*c^3*d^5*x*e + 16*c^3*d^6 + 9*a^2*c*d*x*e^5 + a^3*e^6 - 3*(3*a*c^2*d^2*x^2 - 2*a^2*c*d^2)*e^4 - (c^3*d
^3*x^3 + 36*a*c^2*d^3*x)*e^3 + 6*(c^3*d^4*x^2 - 4*a*c^2*d^4)*e^2)*sqrt(x*e + d)/(x^2*e^6 + 2*d*x*e^5 + d^2*e^4
)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (107) = 214\).
time = 1.55, size = 450, normalized size = 3.91 \begin {gather*} \begin {cases} - \frac {2 a^{3} e^{6}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 a^{2} c d^{2} e^{4}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {18 a^{2} c d e^{5} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 a c^{2} d^{4} e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {72 a c^{2} d^{3} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {18 a c^{2} d^{2} e^{4} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 c^{3} d^{6}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 c^{3} d^{5} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 c^{3} d^{4} e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 c^{3} d^{3} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{3} \sqrt {d} 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)**(11/2),x)

[Out]

Piecewise((-2*a**3*e**6/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 12*a**2*c*d**2*e**4/(3*d*e**4*sqrt
(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 18*a**2*c*d*e**5*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 4
8*a*c**2*d**4*e**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 72*a*c**2*d**3*e**3*x/(3*d*e**4*sqrt(d
+ e*x) + 3*e**5*x*sqrt(d + e*x)) + 18*a*c**2*d**2*e**4*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x))
- 32*c**3*d**6/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 48*c**3*d**5*e*x/(3*d*e**4*sqrt(d + e*x) +
3*e**5*x*sqrt(d + e*x)) - 12*c**3*d**4*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 2*c**3*d*
*3*e**3*x**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)), Ne(e, 0)), (c**3*sqrt(d)*x**4/4, True))

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Giac [A]
time = 2.94, size = 155, normalized size = 1.35 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{8} - 9 \, \sqrt {x e + d} c^{3} d^{4} e^{8} + 9 \, \sqrt {x e + d} a c^{2} d^{2} e^{10}\right )} e^{\left (-12\right )} - \frac {2 \, {\left (9 \, {\left (x e + d\right )} c^{3} d^{5} - c^{3} d^{6} - 18 \, {\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 9 \, {\left (x e + d\right )} a^{2} c d e^{4} - 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} e^{\left (-4\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \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)^(11/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c^3*d^3*e^8 - 9*sqrt(x*e + d)*c^3*d^4*e^8 + 9*sqrt(x*e + d)*a*c^2*d^2*e^10)*e^(-12) - 2/3
*(9*(x*e + d)*c^3*d^5 - c^3*d^6 - 18*(x*e + d)*a*c^2*d^3*e^2 + 3*a*c^2*d^4*e^2 + 9*(x*e + d)*a^2*c*d*e^4 - 3*a
^2*c*d^2*e^4 + a^3*e^6)*e^(-4)/(x*e + d)^(3/2)

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Mupad [B]
time = 0.65, size = 147, normalized size = 1.28 \begin {gather*} -\frac {2\,a^3\,e^6-2\,c^3\,d^6-2\,c^3\,d^3\,{\left (d+e\,x\right )}^3+18\,c^3\,d^4\,{\left (d+e\,x\right )}^2+18\,c^3\,d^5\,\left (d+e\,x\right )+6\,a\,c^2\,d^4\,e^2-6\,a^2\,c\,d^2\,e^4-36\,a\,c^2\,d^3\,e^2\,\left (d+e\,x\right )-18\,a\,c^2\,d^2\,e^2\,{\left (d+e\,x\right )}^2+18\,a^2\,c\,d\,e^4\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \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)^(11/2),x)

[Out]

-(2*a^3*e^6 - 2*c^3*d^6 - 2*c^3*d^3*(d + e*x)^3 + 18*c^3*d^4*(d + e*x)^2 + 18*c^3*d^5*(d + e*x) + 6*a*c^2*d^4*
e^2 - 6*a^2*c*d^2*e^4 - 36*a*c^2*d^3*e^2*(d + e*x) - 18*a*c^2*d^2*e^2*(d + e*x)^2 + 18*a^2*c*d*e^4*(d + e*x))/
(3*e^4*(d + e*x)^(3/2))

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