\(\int \frac {(d+e x)^m}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\) [1096]

Optimal result

Integrand size = 32, antiderivative size = 40 \[ \int \frac {(d+e x)^m}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {(d+e x)^{1+m}}{e m \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, 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 = {658, 32} \[ \int \frac {(d+e x)^m}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {(d+e x)^{m+1}}{e m \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[In]

Int[(d + e*x)^m/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

(d + e*x)^(1 + m)/(e*m*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 658

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \frac {(d+e x)^m}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {(d+e x)^{1+m}}{e m \sqrt {c (d+e x)^2}} \]

[In]

Integrate[(d + e*x)^m/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

(d + e*x)^(1 + m)/(e*m*Sqrt[c*(d + e*x)^2])

Maple [A] (verified)

Time = 2.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {\left (e x +d \right ) \left (e x +d \right )^{m}}{\sqrt {c \left (e x +d \right )^{2}}\, e m}\)	\(31\)
gosper	\(\frac {\left (e x +d \right )^{1+m}}{e m \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}\)	\(39\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^m}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x + d\right )}^{m}}{c e^{2} m x + c d e m} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((d + e*x)**m/sqrt(c*(d + e*x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.42 \[ \int \frac {(d+e x)^m}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {{\left (e x + d\right )}^{m}}{\sqrt {c} e m} \]

[In]

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

[Out]

(e*x + d)^m/(sqrt(c)*e*m)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.62 \[ \int \frac {(d+e x)^m}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {{\left (e x + d\right )}^{m}}{\sqrt {c} e m \mathrm {sgn}\left (e x + d\right )} \]

[In]

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

[Out]

(e*x + d)^m/(sqrt(c)*e*m*sgn(e*x + d))

Mupad [B] (verification not implemented)

Time = 9.66 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int \frac {(d+e x)^m}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx=\frac {{\left (d+e\,x\right )}^m\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c\,e^2\,m\,\left (x+\frac {d}{e}\right )} \]

[In]

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

[Out]

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