\(\int \frac {1}{(d+e x)^3 (c d^2+2 c d e x+c e^2 x^2)^{5/2}} \, dx\) [1088]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {c}{7 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \]

[Out]

-1/7*c/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(7/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {657, 643} \[ \int \frac {1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {c}{7 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \]

[In]

Int[1/((d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]

[Out]

-1/7*c/(e*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(7/2))

Rule 643

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

Rule 657

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

Rubi steps \begin{align*} \text {integral}& = c^2 \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}} \, dx \\ & = -\frac {c}{7 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {c}{7 e \left (c (d+e x)^2\right )^{7/2}} \]

[In]

Integrate[1/((d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]

[Out]

-1/7*c/(e*(c*(d + e*x)^2)^(7/2))

Maple [A] (verified)

Time = 2.60 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
risch	\(-\frac {1}{7 c^{2} \left (e x +d \right )^{6} \sqrt {c \left (e x +d \right )^{2}}\, e}\)	\(27\)
pseudoelliptic	\(-\frac {1}{7 c^{2} \left (e x +d \right )^{6} \sqrt {c \left (e x +d \right )^{2}}\, e}\)	\(27\)
gosper	\(-\frac {1}{7 \left (e x +d \right )^{2} e \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(35\)
default	\(-\frac {1}{7 \left (e x +d \right )^{2} e \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(35\)
trager	\(\frac {\left (e^{6} x^{6}+7 d \,e^{5} x^{5}+21 d^{2} e^{4} x^{4}+35 x^{3} d^{3} e^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{7 c^{3} d^{7} \left (e x +d \right )^{8}}\)	\(101\)

[In]

int(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/7/c^2/(e*x+d)^6/(c*(e*x+d)^2)^(1/2)/e

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (28) = 56\).

Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.34 \[ \int \frac {1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{7 \, {\left (c^{3} e^{9} x^{8} + 8 \, c^{3} d e^{8} x^{7} + 28 \, c^{3} d^{2} e^{7} x^{6} + 56 \, c^{3} d^{3} e^{6} x^{5} + 70 \, c^{3} d^{4} e^{5} x^{4} + 56 \, c^{3} d^{5} e^{4} x^{3} + 28 \, c^{3} d^{6} e^{3} x^{2} + 8 \, c^{3} d^{7} e^{2} x + c^{3} d^{8} e\right )}} \]

[In]

integrate(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/7*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^3*e^9*x^8 + 8*c^3*d*e^8*x^7 + 28*c^3*d^2*e^7*x^6 + 56*c^3*d^3*e^6*
x^5 + 70*c^3*d^4*e^5*x^4 + 56*c^3*d^5*e^4*x^3 + 28*c^3*d^6*e^3*x^2 + 8*c^3*d^7*e^2*x + c^3*d^8*e)

Sympy [F]

\[ \int \frac {1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{3}}\, dx \]

[In]

integrate(1/(e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

Integral(1/((c*(d + e*x)**2)**(5/2)*(d + e*x)**3), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (28) = 56\).

Time = 0.19 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.22 \[ \int \frac {1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{7 \, {\left (c^{\frac {5}{2}} e^{8} x^{7} + 7 \, c^{\frac {5}{2}} d e^{7} x^{6} + 21 \, c^{\frac {5}{2}} d^{2} e^{6} x^{5} + 35 \, c^{\frac {5}{2}} d^{3} e^{5} x^{4} + 35 \, c^{\frac {5}{2}} d^{4} e^{4} x^{3} + 21 \, c^{\frac {5}{2}} d^{5} e^{3} x^{2} + 7 \, c^{\frac {5}{2}} d^{6} e^{2} x + c^{\frac {5}{2}} d^{7} e\right )}} \]

[In]

integrate(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

-1/7/(c^(5/2)*e^8*x^7 + 7*c^(5/2)*d*e^7*x^6 + 21*c^(5/2)*d^2*e^6*x^5 + 35*c^(5/2)*d^3*e^5*x^4 + 35*c^(5/2)*d^4
*e^4*x^3 + 21*c^(5/2)*d^5*e^3*x^2 + 7*c^(5/2)*d^6*e^2*x + c^(5/2)*d^7*e)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{7 \, {\left (e x + d\right )}^{7} c^{\frac {5}{2}} e \mathrm {sgn}\left (e x + d\right )} \]

[In]

integrate(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="giac")

[Out]

-1/7/((e*x + d)^7*c^(5/2)*e*sgn(e*x + d))

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{7\,c^3\,e\,{\left (d+e\,x\right )}^8} \]

[In]

int(1/((d + e*x)^3*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)),x)

[Out]

-(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2)/(7*c^3*e*(d + e*x)^8)