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

Optimal. Leaf size=40 \[ \frac {(d+e x)^{1+m}}{e m \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {658, 32} \begin {gather*} \frac {(d+e x)^{m+1}}{e m \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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) \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*}

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Mathematica [A]
time = 0.02, size = 29, normalized size = 0.72 \begin {gather*} \frac {(d+e x)^{1+m}}{e m \sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [A]
time = 0.59, size = 31, normalized size = 0.78

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

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [A]
time = 0.28, size = 19, normalized size = 0.48 \begin {gather*} \frac {e^{\left (m \log \left (x e + d\right ) - 1\right )}}{\sqrt {c} m} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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^(m*log(x*e + d) - 1)/(sqrt(c)*m)

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Fricas [A]
time = 3.15, size = 46, normalized size = 1.15 \begin {gather*} \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (x e + d\right )}^{m}}{c m x e^{2} + c d m e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{m}}{\sqrt {c \left (d + e x\right )^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]
time = 2.18, size = 26, normalized size = 0.65 \begin {gather*} \frac {{\left (x e + d\right )}^{m} e^{\left (-1\right )}}{\sqrt {c} m \mathrm {sgn}\left (x e + d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

(x*e + d)^m*e^(-1)/(sqrt(c)*m*sgn(x*e + d))

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Mupad [B]
time = 0.49, size = 48, normalized size = 1.20 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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