\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^5} \, dx\) [1927]

Optimal result

Integrand size = 37, antiderivative size = 54 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 \left (c d^2-a e^2\right ) (d+e x)^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {664} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

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

Rule 664

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a +
b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b*e))), 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 + 2*p + 2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 \left (c d^2-a e^2\right ) (d+e x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2}}{5 \left (c d^2-a e^2\right ) (d+e x)^5} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

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

Maple [A] (verified)

Time = 3.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{5 \left (e x +d \right )^{4} \left (e^{2} a -c \,d^{2}\right )}\)	\(58\)
default	\(-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5 e^{5} \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{5}}\)	\(65\)
trager	\(-\frac {2 \left (c^{2} d^{2} x^{2}+2 a x c d e +a^{2} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{5 \left (e x +d \right )^{3} \left (e^{2} a -c \,d^{2}\right )}\)	\(75\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (50) = 100\).

Time = 0.82 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.41 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{5 \, {\left (c d^{5} - a d^{3} e^{2} + {\left (c d^{2} e^{3} - a e^{5}\right )} x^{3} + 3 \, {\left (c d^{3} e^{2} - a d e^{4}\right )} x^{2} + 3 \, {\left (c d^{4} e - a d^{2} e^{3}\right )} x\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**5,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assu
me?` for mor

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 845 vs. \(2 (50) = 100\).

Time = 0.38 (sec) , antiderivative size = 845, normalized size of antiderivative = 15.65 \[ \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)^5} \, dx=-\frac {2}{15} \, {\left (\frac {3 \, \sqrt {c d e} c^{2} d^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c d^{2} e^{3} {\left | e \right |} - a e^{5} {\left | e \right |}} - \frac {\frac {{\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {5}{2}}\right )} c^{2} d^{4} e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}} - \frac {2 \, {\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {5}{2}}\right )} a c d^{2} e^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}} + \frac {{\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {5}{2}}\right )} a^{2} e^{6} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}} - \frac {10 \, {\left (3 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c d e - {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}}\right )} c^{2} d^{3} e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c d^{2} e^{2} - a e^{4}} + \frac {10 \, {\left (3 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c d e - {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}}\right )} a c d e^{3} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c d^{2} e^{2} - a e^{4}} + 15 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{2} d^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c d^{2} e^{3} {\left | e \right |} - a e^{5} {\left | e \right |}}\right )} {\left | e \right |} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

-2/15*(3*sqrt(c*d*e)*c^2*d^2*sgn(1/(e*x + d))*sgn(e)/(c*d^2*e^3*abs(e) - a*e^5*abs(e)) - ((15*sqrt(c*d*e - c*d
^2*e/(e*x + d) + a*e^3/(e*x + d))*c^2*d^2*e^2 - 10*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c*d*e +
3*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2))*c^2*d^4*e^2*sgn(1/(e*x + d))*sgn(e)/(c^2*d^4*e^4 - 2*a
*c*d^2*e^6 + a^2*e^8) - 2*(15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^2*d^2*e^2 - 10*(c*d*e - c*d^
2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c*d*e + 3*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2))*a*c*d^2*
e^4*sgn(1/(e*x + d))*sgn(e)/(c^2*d^4*e^4 - 2*a*c*d^2*e^6 + a^2*e^8) + (15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e
^3/(e*x + d))*c^2*d^2*e^2 - 10*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c*d*e + 3*(c*d*e - c*d^2*e/
(e*x + d) + a*e^3/(e*x + d))^(5/2))*a^2*e^6*sgn(1/(e*x + d))*sgn(e)/(c^2*d^4*e^4 - 2*a*c*d^2*e^6 + a^2*e^8) -
10*(3*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c*d*e - (c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^
(3/2))*c^2*d^3*e*sgn(1/(e*x + d))*sgn(e)/(c*d^2*e^2 - a*e^4) + 10*(3*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e
*x + d))*c*d*e - (c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2))*a*c*d*e^3*sgn(1/(e*x + d))*sgn(e)/(c*d^2
*e^2 - a*e^4) + 15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^2*d^2*sgn(1/(e*x + d))*sgn(e))/(c*d^2*e
^3*abs(e) - a*e^5*abs(e)))*abs(e)

Mupad [B] (verification not implemented)

Time = 11.36 (sec) , antiderivative size = 901, normalized size of antiderivative = 16.69 \[ \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)^5} \, dx=\frac {\left (\frac {d\,\left (\frac {4\,c^3\,d^4}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {2\,c^2\,d^2\,\left (5\,a\,e^2-c\,d^2\right )}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )}{e}+\frac {4\,a^2\,c\,d\,e^4+2\,a\,c^2\,d^3\,e^2-2\,c^3\,d^5}{5\,e\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {d\,\left (\frac {40\,c^4\,d^5-56\,a\,c^3\,d^3\,e^2}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}+\frac {8\,c^4\,d^5}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}\right )}{e}+\frac {8\,a\,c^2\,d^2\,\left (6\,a\,e^2-5\,c\,d^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {2\,a^2\,e^3}{5\,a\,e^3-5\,c\,d^2\,e}+\frac {d\,\left (\frac {2\,c^2\,d^3}{5\,a\,e^3-5\,c\,d^2\,e}-\frac {4\,a\,c\,d\,e^2}{5\,a\,e^3-5\,c\,d^2\,e}\right )}{e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {10\,c^3\,d^4-22\,a\,c^2\,d^2\,e^2}{15\,e^2\,{\left (a\,e^2-c\,d^2\right )}^2}+\frac {4\,c^3\,d^4}{5\,e^2\,{\left (a\,e^2-c\,d^2\right )}^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {d\,\left (\frac {12\,c^3\,d^4-20\,a\,c^2\,d^2\,e^2}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}+\frac {4\,c^3\,d^4}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )}{e}+\frac {4\,a\,c\,d\,e\,\left (4\,a\,e^2-3\,c\,d^2\right )}{5\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {d\,\left (\frac {8\,c^4\,d^5}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}-\frac {4\,c^3\,d^3\,\left (11\,a\,e^2-7\,c\,d^2\right )}{15\,e\,{\left (a\,e^2-c\,d^2\right )}^3}\right )}{e}+\frac {20\,a^2\,c^2\,d^2\,e^4+4\,a\,c^3\,d^4\,e^2-16\,c^4\,d^6}{15\,e^2\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(d + e*x)^5,x)

[Out]

(((d*((4*c^3*d^4)/(5*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) - (2*c^2*d^2*(5*a*e^2 - c*d^2))/(5*(a*e^2 - c*d^2)
*(3*a*e^3 - 3*c*d^2*e))))/e + (2*a*c^2*d^3*e^2 - 2*c^3*d^5 + 4*a^2*c*d*e^4)/(5*e*(a*e^2 - c*d^2)*(3*a*e^3 - 3*
c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (((d*((40*c^4*d^5 - 56*a*c^3*d^3*e^2)/
(15*e*(a*e^2 - c*d^2)^3) + (8*c^4*d^5)/(15*e*(a*e^2 - c*d^2)^3)))/e + (8*a*c^2*d^2*(6*a*e^2 - 5*c*d^2))/(15*(a
*e^2 - c*d^2)^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((2*a^2*e^3)/(5*a*e^3 - 5*c*d^2*
e) + (d*((2*c^2*d^3)/(5*a*e^3 - 5*c*d^2*e) - (4*a*c*d*e^2)/(5*a*e^3 - 5*c*d^2*e)))/e)*(x*(a*e^2 + c*d^2) + a*d
*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - (((10*c^3*d^4 - 22*a*c^2*d^2*e^2)/(15*e^2*(a*e^2 - c*d^2)^2) + (4*c^3*d^4
)/(5*e^2*(a*e^2 - c*d^2)^2))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((12*c^3*d^4 - 20
*a*c^2*d^2*e^2)/(5*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) + (4*c^3*d^4)/(5*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*
e))))/e + (4*a*c*d*e*(4*a*e^2 - 3*c*d^2))/(5*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*
e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + (((d*((8*c^4*d^5)/(15*e*(a*e^2 - c*d^2)^3) - (4*c^3*d^3*(11*a*e^2 - 7*c*d^
2))/(15*e*(a*e^2 - c*d^2)^3)))/e + (4*a*c^3*d^4*e^2 - 16*c^4*d^6 + 20*a^2*c^2*d^2*e^4)/(15*e^2*(a*e^2 - c*d^2)
^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)