3.11.94 \(\int (d+e x)^m (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx\) [1094]

Optimal. Leaf size=42 \[ \frac {(d+e x)^{1+m} \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{e (4+m)} \]

[Out]

(e*x+d)^(1+m)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/e/(4+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 {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} (d+e x)^{m+1}}{e (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))/(e*(4 + 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 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx &=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \int (d+e x)^{3+m} \, dx}{(d+e x)^3}\\ &=\frac {(d+e x)^{1+m} \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{e (4+m)}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 31, normalized size = 0.74 \begin {gather*} \frac {(d+e x)^{1+m} \left (c (d+e x)^2\right )^{3/2}}{e (4+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(c*(d + e*x)^2)^(3/2))/(e*(4 + m))

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Maple [A]
time = 0.59, size = 41, normalized size = 0.98

method	result	size
gosper	\(\frac {\left (e x +d \right )^{1+m} \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{e \left (4+m \right )}\)	\(41\)
risch	\(\frac {c \sqrt {\left (e x +d \right )^{2} c}\, \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right ) \left (e x +d \right )^{m}}{\left (e x +d \right ) e \left (4+m \right )}\)	\(74\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

(e*x+d)^(1+m)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/e/(4+m)

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Maxima [A]
time = 0.29, size = 70, normalized size = 1.67 \begin {gather*} \frac {{\left (c^{\frac {3}{2}} x^{4} e^{4} + 4 \, c^{\frac {3}{2}} d x^{3} e^{3} + 6 \, c^{\frac {3}{2}} d^{2} x^{2} e^{2} + 4 \, c^{\frac {3}{2}} d^{3} x e + c^{\frac {3}{2}} d^{4}\right )} e^{\left (m \log \left (x e + d\right ) - 1\right )}}{m + 4} \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)^(3/2),x, algorithm="maxima")

[Out]

(c^(3/2)*x^4*e^4 + 4*c^(3/2)*d*x^3*e^3 + 6*c^(3/2)*d^2*x^2*e^2 + 4*c^(3/2)*d^3*x*e + c^(3/2)*d^4)*e^(m*log(x*e
+ d) - 1)/(m + 4)

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Fricas [A]
time = 3.48, size = 69, normalized size = 1.64 \begin {gather*} \frac {{\left (c x^{3} e^{3} + 3 \, c d x^{2} e^{2} + 3 \, c d^{2} x e + c d^{3}\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (x e + d\right )}^{m} e^{\left (-1\right )}}{m + 4} \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)^(3/2),x, algorithm="fricas")

[Out]

(c*x^3*e^3 + 3*c*d*x^2*e^2 + 3*c*d^2*x*e + c*d^3)*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)*(x*e + d)^m*e^(-1)/(m +
4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c \left (d + e x\right )^{2}\right )^{\frac {3}{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)**(3/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (40) = 80\).
time = 2.85, size = 143, normalized size = 3.40 \begin {gather*} \frac {{\left (c x^{3} e^{\left (m \log \left (x e + d\right ) + \log \left (x e + d\right ) + 3\right )} \mathrm {sgn}\left (x e + d\right ) + 3 \, c d x^{2} e^{\left (m \log \left (x e + d\right ) + \log \left (x e + d\right ) + 2\right )} \mathrm {sgn}\left (x e + d\right ) + 3 \, c d^{2} x e^{\left (m \log \left (x e + d\right ) + \log \left (x e + d\right ) + 1\right )} \mathrm {sgn}\left (x e + d\right ) + c d^{3} e^{\left (m \log \left (x e + d\right ) + \log \left (x e + d\right )\right )} \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c}}{m e + 4 \, 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)^(3/2),x, algorithm="giac")

[Out]

(c*x^3*e^(m*log(x*e + d) + log(x*e + d) + 3)*sgn(x*e + d) + 3*c*d*x^2*e^(m*log(x*e + d) + log(x*e + d) + 2)*sg
n(x*e + d) + 3*c*d^2*x*e^(m*log(x*e + d) + log(x*e + d) + 1)*sgn(x*e + d) + c*d^3*e^(m*log(x*e + d) + log(x*e
+ d))*sgn(x*e + d))*sqrt(c)/(m*e + 4*e)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (d+e\,x\right )}^m\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2} \,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)^(3/2),x)

[Out]

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

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