3.9.63 \(\int (d+e x)^2 (c d^2+2 c d e x+c e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=39 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e} \]

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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 )^{7/2}}{8 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(7/2))/(8*c*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 (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx &=\frac {\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2} \, dx}{c}\\ &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.72 \begin {gather*} \frac {(d+e x) \left (c (d+e x)^2\right )^{7/2}}{8 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*(c*(d + e*x)^2)^(7/2))/(8*c*e)

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IntegrateAlgebraic [F]  time = 0.99, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[IntegrateAlgebraic][(d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2), x]

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fricas [B]  time = 0.39, size = 131, normalized size = 3.36 \begin {gather*} \frac {{\left (c^{2} e^{7} x^{8} + 8 \, c^{2} d e^{6} x^{7} + 28 \, c^{2} d^{2} e^{5} x^{6} + 56 \, c^{2} d^{3} e^{4} x^{5} + 70 \, c^{2} d^{4} e^{3} x^{4} + 56 \, c^{2} d^{5} e^{2} x^{3} + 28 \, c^{2} d^{6} e x^{2} + 8 \, c^{2} d^{7} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{8 \, {\left (e x + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(c^2*e^7*x^8 + 8*c^2*d*e^6*x^7 + 28*c^2*d^2*e^5*x^6 + 56*c^2*d^3*e^4*x^5 + 70*c^2*d^4*e^3*x^4 + 56*c^2*d^5
*e^2*x^3 + 28*c^2*d^6*e*x^2 + 8*c^2*d^7*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)

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giac [B]  time = 0.31, size = 115, normalized size = 2.95 \begin {gather*} \frac {1}{8} \, {\left (c^{2} d^{7} e^{\left (-1\right )} + {\left (7 \, c^{2} d^{6} + {\left (21 \, c^{2} d^{5} e + {\left (35 \, c^{2} d^{4} e^{2} + {\left (35 \, c^{2} d^{3} e^{3} + {\left (21 \, c^{2} d^{2} e^{4} + {\left (c^{2} x e^{6} + 7 \, c^{2} d e^{5}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(c^2*d^7*e^(-1) + (7*c^2*d^6 + (21*c^2*d^5*e + (35*c^2*d^4*e^2 + (35*c^2*d^3*e^3 + (21*c^2*d^2*e^4 + (c^2*
x*e^6 + 7*c^2*d*e^5)*x)*x)*x)*x)*x)*x)*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)

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maple [B]  time = 0.04, size = 106, normalized size = 2.72 \begin {gather*} \frac {\left (e^{7} x^{7}+8 d \,e^{6} x^{6}+28 e^{5} x^{5} d^{2}+56 e^{4} x^{4} d^{3}+70 d^{4} e^{3} x^{3}+56 d^{5} e^{2} x^{2}+28 d^{6} e x +8 d^{7}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} x}{8 \left (e x +d \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x*(e^7*x^7+8*d*e^6*x^6+28*d^2*e^5*x^5+56*d^3*e^4*x^4+70*d^4*e^3*x^3+56*d^5*e^2*x^2+28*d^6*e*x+8*d^7)*(c*e^
2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^5

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maxima [A]  time = 1.48, size = 60, normalized size = 1.54 \begin {gather*} \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {7}{2}} x}{8 \, c} + \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {7}{2}} d}{8 \, c e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(7/2)*x/c + 1/8*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(7/2)*d/(c*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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