\(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^6} \, dx\) [1059]

Optimal result

Integrand size = 32, antiderivative size = 42 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {c^3 (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

c^3*(e*x+d)*ln(e*x+d)/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {656, 622, 31} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {c^3 (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^6,x]

[Out]

(c^3*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 622

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 656

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

Rubi steps \begin{align*} \text {integral}& = c^3 \int \frac {1}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx \\ & = \frac {\left (c^3 \left (c d e+c e^2 x\right )\right ) \int \frac {1}{c d e+c e^2 x} \, dx}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ & = \frac {c^3 (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {c^3 (d+e x) \log (d+e x)}{e \sqrt {c (d+e x)^2}} \]

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^6,x]

[Out]

(c^3*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*(d + e*x)^2])

Maple [A] (verified)

Time = 2.94 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {c^{2} \sqrt {c \left (e x +d \right )^{2}}\, \ln \left (e x +d \right )}{\left (e x +d \right ) e}\)	\(32\)
default	\(\frac {\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}} \ln \left (e x +d \right )}{\left (e x +d \right )^{5} e}\)	\(40\)

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c^{2} \log \left (e x + d\right )}{e^{2} x + d e} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*c^2*log(e*x + d)/(e^2*x + d*e)

Sympy [F]

\[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \]

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2)/(e*x+d)**6,x)

[Out]

Integral((c*(d + e*x)**2)**(5/2)/(d + e*x)**6, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^6,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.48 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {c^{\frac {5}{2}} \log \left ({\left | e x + d \right |}\right ) \mathrm {sgn}\left (e x + d\right )}{e} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

c^(5/2)*log(abs(e*x + d))*sgn(e*x + d)/e

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \]

[In]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^6,x)

[Out]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^6, x)