3.11.56 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^3} \, dx\) [1056]

Optimal. Leaf size=32 \[ \frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 32, 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 = {657, 643} \begin {gather*} \frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

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

Rule 643

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*((a + b*x + c*x^2)^(p +
1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 657

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c,
0] &&  !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx &=c^2 \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx\\ &=\frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.66 \begin {gather*} \frac {c \left (c (d+e x)^2\right )^{3/2}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

(c*(c*(d + e*x)^2)^(3/2))/(3*e)

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Maple [A]
time = 0.58, size = 35, normalized size = 1.09

method	result	size
risch	\(\frac {c^{2} \left (e x +d \right )^{2} \sqrt {\left (e x +d \right )^{2} c}}{3 e}\)	\(27\)
default	\(\frac {\left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{3 \left (e x +d \right )^{2} e}\)	\(35\)
gosper	\(\frac {x \left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{3 \left (e x +d \right )^{5}}\)	\(51\)
trager	\(\frac {c^{2} x \left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{3 e x +3 d}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/3*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^2/e

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).
time = 2.57, size = 62, normalized size = 1.94 \begin {gather*} \frac {{\left (c^{2} x^{3} e^{2} + 3 \, c^{2} d x^{2} e + 3 \, c^{2} d^{2} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{3 \, {\left (x e + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2)/(e*x+d)**3,x)

[Out]

Integral((c*(d + e*x)**2)**(5/2)/(d + e*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (27) = 54\).
time = 0.99, size = 56, normalized size = 1.75 \begin {gather*} \frac {1}{3} \, {\left (c^{2} x^{3} e^{2} \mathrm {sgn}\left (x e + d\right ) + 3 \, c^{2} d x^{2} e \mathrm {sgn}\left (x e + d\right ) + 3 \, c^{2} d^{2} x \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

1/3*(c^2*x^3*e^2*sgn(x*e + d) + 3*c^2*d*x^2*e*sgn(x*e + d) + 3*c^2*d^2*x*sgn(x*e + d))*sqrt(c)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^3,x)

[Out]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^3, x)

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