3.9.74 \(\int \frac {(d+e x)^4}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\)

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

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Rubi [A]  time = 0.02, 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 = {642, 609} \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 609

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 642

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*}

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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])

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IntegrateAlgebraic [F]  time = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^4}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 66, normalized size = 1.69 \begin {gather*} \frac {{\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (c e x + 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*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c*e*x + c*d)

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giac [A]  time = 0.72, size = 63, normalized size = 1.62 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (\frac {d^{3} e^{\left (-1\right )}}{c} + {\left (x {\left (\frac {x e^{2}}{c} + \frac {3 \, d e}{c}\right )} + \frac {3 \, d^{2}}{c}\right )} x\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^2*e^2 + 2*c*d*x*e + c*d^2)*(d^3*e^(-1)/c + (x*(x*e^2/c + 3*d*e/c) + 3*d^2/c)*x)

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maple [A]  time = 0.05, size = 60, normalized size = 1.54 \begin {gather*} \frac {\left (e^{3} x^{3}+4 e^{2} x^{2} d +6 d^{2} x e +4 d^{3}\right ) \left (e x +d \right ) x}{4 \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}} \end {gather*}

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)

[Out]

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

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maxima [B]  time = 1.49, size = 120, normalized size = 3.08 \begin {gather*} \frac {3 \, d^{2} e x^{2}}{4 \, \sqrt {c}} + \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} e^{2} x^{3}}{4 \, c} - \frac {3 \, d^{3} x}{2 \, \sqrt {c}} + \frac {3 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d e x^{2}}{4 \, c} + \frac {5 \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d^{3}}{2 \, c e} \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*e*x^2/sqrt(c) + 1/4*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*e^2*x^3/c - 3/2*d^3*x/sqrt(c) + 3/4*sqrt(c*e^2
*x^2 + 2*c*d*e*x + c*d^2)*d*e*x^2/c + 5/2*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*d^3/(c*e)

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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)

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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)

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