3.20.99 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{13/2}} \, dx\) [1999]

Optimal. Leaf size=113 \[ \frac {2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}-\frac {2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac {6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 c^3 d^3 \sqrt {d+e x}}{e^4} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 113, 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 \left (c d^2-a e^2\right )}{e^4 \sqrt {d+e x}}-\frac {2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac {2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}+\frac {2 c^3 d^3 \sqrt {d+e x}}{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)^(13/2),x]

[Out]

(2*(c*d^2 - a*e^2)^3)/(5*e^4*(d + e*x)^(5/2)) - (2*c*d*(c*d^2 - a*e^2)^2)/(e^4*(d + e*x)^(3/2)) + (6*c^2*d^2*(
c*d^2 - a*e^2))/(e^4*Sqrt[d + e*x]) + (2*c^3*d^3*Sqrt[d + e*x])/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)^{13/2}} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^{7/2}}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^{5/2}}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^{3/2}}+\frac {c^3 d^3}{e^3 \sqrt {d+e x}}\right ) \, dx\\ &=\frac {2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}-\frac {2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac {6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 c^3 d^3 \sqrt {d+e x}}{e^4}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 109, normalized size = 0.96 \begin {gather*} -\frac {2 \left (a^3 e^6+a^2 c d e^4 (2 d+5 e x)+a c^2 d^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-c^3 d^3 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )}{5 e^4 (d+e x)^{5/2}} \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)^(13/2),x]

[Out]

(-2*(a^3*e^6 + a^2*c*d*e^4*(2*d + 5*e*x) + a*c^2*d^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - c^3*d^3*(16*d^3 + 4
0*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)))/(5*e^4*(d + e*x)^(5/2))

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Maple [A]
time = 0.71, size = 133, normalized size = 1.18

method	result	size
risch	\(\frac {2 c^{3} d^{3} \sqrt {e x +d}}{e^{4}}-\frac {2 \left (15 e^{2} x^{2} c^{2} d^{2}+5 a c d \,e^{3} x +25 c^{2} d^{3} e x +a^{2} e^{4}+3 a c \,d^{2} e^{2}+11 c^{2} d^{4}\right ) \left (e^{2} a -c \,d^{2}\right )}{5 e^{4} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d x e +d^{2}\right )}\)	\(121\)
gosper	\(-\frac {2 \left (-5 c^{3} d^{3} e^{3} x^{3}+15 a \,c^{2} d^{2} e^{4} x^{2}-30 c^{3} d^{4} e^{2} x^{2}+5 a^{2} c d \,e^{5} x +20 a \,c^{2} d^{3} e^{3} x -40 c^{3} d^{5} e x +e^{6} a^{3}+2 e^{4} d^{2} a^{2} c +8 d^{4} e^{2} c^{2} a -16 d^{6} c^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{4}}\)	\(130\)
trager	\(-\frac {2 \left (-5 c^{3} d^{3} e^{3} x^{3}+15 a \,c^{2} d^{2} e^{4} x^{2}-30 c^{3} d^{4} e^{2} x^{2}+5 a^{2} c d \,e^{5} x +20 a \,c^{2} d^{3} e^{3} x -40 c^{3} d^{5} e x +e^{6} a^{3}+2 e^{4} d^{2} a^{2} c +8 d^{4} e^{2} c^{2} a -16 d^{6} c^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{4}}\)	\(130\)
derivativedivides	\(\frac {2 c^{3} d^{3} \sqrt {e x +d}-\frac {2 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\left (e x +d \right )^{\frac {3}{2}}}-\frac {6 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{4}}\)	\(133\)
default	\(\frac {2 c^{3} d^{3} \sqrt {e x +d}-\frac {2 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\left (e x +d \right )^{\frac {3}{2}}}-\frac {6 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{4}}\)	\(133\)

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)^(13/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 135, normalized size = 1.19 \begin {gather*} \frac {2}{5} \, {\left (5 \, \sqrt {x e + d} c^{3} d^{3} 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} + 15 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{2} - 5 \, {\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 {5}{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)^(13/2),x, algorithm="maxima")

[Out]

2/5*(5*sqrt(x*e + d)*c^3*d^3*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 + 15*(c^3*d^4 - a
*c^2*d^2*e^2)*(x*e + d)^2 - 5*(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)^(5/2))*e^(
-1)

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Fricas [A]
time = 2.41, size = 154, normalized size = 1.36 \begin {gather*} \frac {2 \, {\left (40 \, c^{3} d^{5} x e + 16 \, c^{3} d^{6} - 5 \, a^{2} c d x e^{5} - a^{3} e^{6} - {\left (15 \, a c^{2} d^{2} x^{2} + 2 \, a^{2} c d^{2}\right )} e^{4} + 5 \, {\left (c^{3} d^{3} x^{3} - 4 \, a c^{2} d^{3} x\right )} e^{3} + 2 \, {\left (15 \, c^{3} d^{4} x^{2} - 4 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{5 \, {\left (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + d^{3} 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)^(13/2),x, algorithm="fricas")

[Out]

2/5*(40*c^3*d^5*x*e + 16*c^3*d^6 - 5*a^2*c*d*x*e^5 - a^3*e^6 - (15*a*c^2*d^2*x^2 + 2*a^2*c*d^2)*e^4 + 5*(c^3*d
^3*x^3 - 4*a*c^2*d^3*x)*e^3 + 2*(15*c^3*d^4*x^2 - 4*a*c^2*d^4)*e^2)*sqrt(x*e + d)/(x^3*e^7 + 3*d*x^2*e^6 + 3*d
^2*x*e^5 + d^3*e^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (105) = 210\).
time = 2.22, size = 654, normalized size = 5.79 \begin {gather*} \begin {cases} - \frac {2 a^{3} e^{6}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {4 a^{2} c d^{2} e^{4}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {10 a^{2} c d e^{5} x}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {16 a c^{2} d^{4} e^{2}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {40 a c^{2} d^{3} e^{3} x}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {30 a c^{2} d^{2} e^{4} x^{2}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {32 c^{3} d^{6}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {80 c^{3} d^{5} e x}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {60 c^{3} d^{4} e^{2} x^{2}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {10 c^{3} d^{3} e^{3} x^{3}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{3} x^{4}}{4 \sqrt {d}} & \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)**(13/2),x)

[Out]

Piecewise((-2*a**3*e**6/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) -
4*a**2*c*d**2*e**4/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) - 10*a*
*2*c*d*e**5*x/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) - 16*a*c**2*
d**4*e**2/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) - 40*a*c**2*d**3
*e**3*x/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) - 30*a*c**2*d**2*e
**4*x**2/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) + 32*c**3*d**6/(5
*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) + 80*c**3*d**5*e*x/(5*d**2*e
**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) + 60*c**3*d**4*e**2*x**2/(5*d**2*e*
*4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) + 10*c**3*d**3*e**3*x**3/(5*d**2*e**
4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)), Ne(e, 0)), (c**3*x**4/(4*sqrt(d)), T
rue))

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Giac [A]
time = 1.10, size = 149, normalized size = 1.32 \begin {gather*} 2 \, \sqrt {x e + d} c^{3} d^{3} e^{\left (-4\right )} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} c^{3} d^{4} - 5 \, {\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} - 15 \, {\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} + 10 \, {\left (x e + d\right )} a c^{2} d^{3} e^{2} - 3 \, a c^{2} d^{4} e^{2} - 5 \, {\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 )}}{5 \, {\left (x e + d\right )}^{\frac {5}{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)^(13/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*c^3*d^3*e^(-4) + 2/5*(15*(x*e + d)^2*c^3*d^4 - 5*(x*e + d)*c^3*d^5 + c^3*d^6 - 15*(x*e + d)^2*
a*c^2*d^2*e^2 + 10*(x*e + d)*a*c^2*d^3*e^2 - 3*a*c^2*d^4*e^2 - 5*(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)^(5/2)

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Mupad [B]
time = 0.65, size = 129, normalized size = 1.14 \begin {gather*} -\frac {2\,\left (a^3\,e^6+2\,a^2\,c\,d^2\,e^4+5\,a^2\,c\,d\,e^5\,x+8\,a\,c^2\,d^4\,e^2+20\,a\,c^2\,d^3\,e^3\,x+15\,a\,c^2\,d^2\,e^4\,x^2-16\,c^3\,d^6-40\,c^3\,d^5\,e\,x-30\,c^3\,d^4\,e^2\,x^2-5\,c^3\,d^3\,e^3\,x^3\right )}{5\,e^4\,{\left (d+e\,x\right )}^{5/2}} \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)^(13/2),x)

[Out]

-(2*(a^3*e^6 - 16*c^3*d^6 + 8*a*c^2*d^4*e^2 + 2*a^2*c*d^2*e^4 - 30*c^3*d^4*e^2*x^2 - 5*c^3*d^3*e^3*x^3 - 40*c^
3*d^5*e*x + 5*a^2*c*d*e^5*x + 20*a*c^2*d^3*e^3*x + 15*a*c^2*d^2*e^4*x^2))/(5*e^4*(d + e*x)^(5/2))

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