3.1094 \(\int (d+e x)^m (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0189694, antiderivative size = 42, normalized size of antiderivative = 1., 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 = {644, 32} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} (d+e x)^{m+1}}{e (m+4)} \]

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 644

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]

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]

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*}

Mathematica [A]  time = 0.0295078, size = 31, normalized size = 0.74 \[ \frac{\left (c (d+e x)^2\right )^{3/2} (d+e x)^{m+1}}{e (m+4)} \]

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 = 0.038, size = 41, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m}}{e \left ( 4+m \right ) } \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

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)

[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 = 1.18199, size = 95, normalized size = 2.26 \begin{align*} \frac{{\left (c^{\frac{3}{2}} e^{4} x^{4} + 4 \, c^{\frac{3}{2}} d e^{3} x^{3} + 6 \, c^{\frac{3}{2}} d^{2} e^{2} x^{2} + 4 \, c^{\frac{3}{2}} d^{3} e x + c^{\frac{3}{2}} d^{4}\right )}{\left (e x + d\right )}^{m}}{e{\left (m + 4\right )}} \end{align*}

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

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Fricas [A]  time = 2.55927, size = 150, normalized size = 3.57 \begin{align*} \frac{{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x + d\right )}^{m}}{e m + 4 \, e} \end{align*}

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{m}\, dx \end{align*}

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]  time = 1.27467, size = 140, normalized size = 3.33 \begin{align*} \frac{{\left (x e + d\right )}^{m} c^{\frac{3}{2}} x^{4} e^{4} + 4 \,{\left (x e + d\right )}^{m} c^{\frac{3}{2}} d x^{3} e^{3} + 6 \,{\left (x e + d\right )}^{m} c^{\frac{3}{2}} d^{2} x^{2} e^{2} + 4 \,{\left (x e + d\right )}^{m} c^{\frac{3}{2}} d^{3} x e +{\left (x e + d\right )}^{m} c^{\frac{3}{2}} d^{4}}{m e + 4 \, e} \end{align*}

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]

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