3.11.62 \(\int \frac {(d+e x)^4}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\) [1062]

Optimal. Leaf size=39 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c^2 e} \]

[Out]

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 27, normalized size = 0.69 \begin {gather*} \frac {(d+e x)^5}{4 e \sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

(d + e*x)^5/(4*e*Sqrt[c*(d + e*x)^2])

________________________________________________________________________________________

Maple [A]
time = 0.59, size = 35, normalized size = 0.90

method	result	size
risch	\(\frac {\left (e x +d \right )^{5}}{4 \sqrt {\left (e x +d \right )^{2} c}\, e}\)	\(24\)
default	\(\frac {\left (e x +d \right )^{5}}{4 \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}\, e}\)	\(35\)
gosper	\(\frac {x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \left (e x +d \right )}{4 \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\)	\(60\)
trager	\(\frac {x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{4 c \left (e x +d \right )}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)/e

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (35) = 70\).
time = 0.30, size = 120, normalized size = 3.08 \begin {gather*} \frac {3 \, d^{2} x^{2} e}{4 \, \sqrt {c}} - \frac {3 \, d^{3} x}{2 \, \sqrt {c}} + \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} x^{3} e^{2}}{4 \, c} + \frac {3 \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} d x^{2} e}{4 \, c} + \frac {5 \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} d^{3} e^{\left (-1\right )}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

3/4*d^2*x^2*e/sqrt(c) - 3/2*d^3*x/sqrt(c) + 1/4*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)*x^3*e^2/c + 3/4*sqrt(c*x^2
*e^2 + 2*c*d*x*e + c*d^2)*d*x^2*e/c + 5/2*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)*d^3*e^(-1)/c

________________________________________________________________________________________

Fricas [A]
time = 3.31, size = 66, normalized size = 1.69 \begin {gather*} \frac {{\left (x^{4} e^{3} + 4 \, d x^{3} e^{2} + 6 \, d^{2} x^{2} e + 4 \, d^{3} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{4 \, {\left (c x e + c d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

1/4*(x^4*e^3 + 4*d*x^3*e^2 + 6*d^2*x^2*e + 4*d^3*x)*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)/(c*x*e + c*d)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{4}}{\sqrt {c \left (d + e x\right )^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.17, size = 57, normalized size = 1.46 \begin {gather*} \frac {\sqrt {c} x^{4} e^{3} + 4 \, \sqrt {c} d x^{3} e^{2} + 6 \, \sqrt {c} d^{2} x^{2} e + 4 \, \sqrt {c} d^{3} x}{4 \, c \mathrm {sgn}\left (x e + d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="giac")

[Out]

1/4*(sqrt(c)*x^4*e^3 + 4*sqrt(c)*d*x^3*e^2 + 6*sqrt(c)*d^2*x^2*e + 4*sqrt(c)*d^3*x)/(c*sgn(x*e + d))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^4}{\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)^4/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2),x)

[Out]

int((d + e*x)^4/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2), x)

________________________________________________________________________________________