3.11.57 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^4} \, dx\) [1057]

Optimal. Leaf size=39 \[ \frac {c^2 (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 39, 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 = {656, 623} \begin {gather*} \frac {c^2 (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 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)^4,x]

[Out]

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

Rule 623

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1)
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rule 656

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^m/c^(m/2), Int[(a +
b*x + c*x^2)^(p + m/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/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)^4} \, dx &=c^2 \int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx\\ &=\frac {c^2 (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 33, normalized size = 0.85 \begin {gather*} \frac {c^3 x (d+e x) (2 d+e x)}{2 \sqrt {c (d+e x)^2}} \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)^4,x]

[Out]

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

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Maple [A]
time = 0.58, size = 40, normalized size = 1.03

method	result	size
gosper	\(\frac {x \left (e x +2 d \right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{2 \left (e x +d \right )^{5}}\)	\(40\)
default	\(\frac {x \left (e x +2 d \right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{2 \left (e x +d \right )^{5}}\)	\(40\)
trager	\(\frac {c^{2} x \left (e x +2 d \right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{2 e x +2 d}\)	\(43\)
risch	\(\frac {c^{2} \sqrt {\left (e x +d \right )^{2} c}\, e \,x^{2}}{2 e x +2 d}+\frac {c^{2} \sqrt {\left (e x +d \right )^{2} c}\, d x}{e x +d}\)	\(53\)

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)^4,x,method=_RETURNVERBOSE)

[Out]

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

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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)^4,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 [A]
time = 2.42, size = 49, normalized size = 1.26 \begin {gather*} \frac {{\left (c^{2} x^{2} e + 2 \, c^{2} d x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{2 \, {\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)^4,x, algorithm="fricas")

[Out]

1/2*(c^2*x^2*e + 2*c^2*d*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 )^{4}}\, 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)**4,x)

[Out]

Integral((c*(d + e*x)**2)**(5/2)/(d + e*x)**4, x)

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Giac [A]
time = 0.73, size = 36, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, {\left (c^{2} x^{2} e \mathrm {sgn}\left (x e + d\right ) + 2 \, c^{2} d 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)^4,x, algorithm="giac")

[Out]

1/2*(c^2*x^2*e*sgn(x*e + d) + 2*c^2*d*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 )}^4} \,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)^4,x)

[Out]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^4, x)

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