3.11.95 \(\int (d+e x)^m \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx\) [1095]

Optimal. Leaf size=42 \[ \frac {(d+e x)^{1+m} \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e (2+m)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 42, 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 {\sqrt {c d^2+2 c d e x+c e^2 x^2} (d+e x)^{m+1}}{e (m+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)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])/(e*(2 + m))

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

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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.74 \begin {gather*} \frac {(d+e x)^{1+m} \sqrt {c (d+e x)^2}}{e (2+m)} \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)*Sqrt[c*(d + e*x)^2])/(e*(2 + m))

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Maple [A]
time = 0.54, size = 41, normalized size = 0.98

method	result	size
gosper	\(\frac {\left (e x +d \right )^{1+m} \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{e \left (2+m \right )}\)	\(41\)
risch	\(\frac {\sqrt {\left (e x +d \right )^{2} c}\, \left (e^{2} x^{2}+2 d x e +d^{2}\right ) \left (e x +d \right )^{m}}{\left (e x +d \right ) e \left (2+m \right )}\)	\(51\)

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]

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

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Maxima [A]
time = 0.28, size = 44, normalized size = 1.05 \begin {gather*} \frac {{\left (\sqrt {c} x^{2} e^{2} + 2 \, \sqrt {c} d x e + \sqrt {c} d^{2}\right )} e^{\left (m \log \left (x e + d\right ) - 1\right )}}{m + 2} \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]

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

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Fricas [A]
time = 2.31, size = 44, normalized size = 1.05 \begin {gather*} \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (x e + d\right )} {\left (x e + d\right )}^{m} e^{\left (-1\right )}}{m + 2} \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)*(x*e + d)^m*e^(-1)/(m + 2)

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

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Giac [A]
time = 1.39, size = 28, normalized size = 0.67 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 2} \sqrt {c} e^{\left (-1\right )} \mathrm {sgn}\left (x e + d\right )}{m + 2} \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 + 2)*sqrt(c)*e^(-1)*sgn(x*e + d)/(m + 2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (d+e\,x\right )}^m\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2} \,d x \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]

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

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