3.2039 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=109 \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}} \]

[Out]

(4*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*c^2*d^2*(d + e*x)^(5/2)) + (2*(a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d*(d + e*x)^(3/2))

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Rubi [A]  time = 0.0610188, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*c^2*d^2*(d + e*x)^(5/2)) + (2*(a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d*(d + e*x)^(3/2))

Rule 656

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

Rule 648

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

Rubi steps

\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}+\frac{\left (2 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{7 d}\\ &=\frac{4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0630417, size = 55, normalized size = 0.5 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (c d (7 d+5 e x)-2 a e^2\right )}{35 c^2 d^2 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-2*a*e^2 + c*d*(7*d + 5*e*x)))/(35*c^2*d^2*(d + e*x)^(5/2))

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Maple [A]  time = 0.043, size = 69, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -5\,cdex+2\,a{e}^{2}-7\,c{d}^{2} \right ) }{35\,{c}^{2}{d}^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(c*d*x+a*e)*(-5*c*d*e*x+2*a*e^2-7*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)/c^2/d^2/(e*x+d)^(3/2)

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Maxima [A]  time = 1.06772, size = 132, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} e x^{3} + 7 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4} +{\left (7 \, c^{3} d^{4} + 8 \, a c^{2} d^{2} e^{2}\right )} x^{2} +{\left (14 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x\right )} \sqrt{c d x + a e}}{35 \, c^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/35*(5*c^3*d^3*e*x^3 + 7*a^2*c*d^2*e^2 - 2*a^3*e^4 + (7*c^3*d^4 + 8*a*c^2*d^2*e^2)*x^2 + (14*a*c^2*d^3*e + a^
2*c*d*e^3)*x)*sqrt(c*d*x + a*e)/(c^2*d^2)

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Fricas [A]  time = 1.82388, size = 279, normalized size = 2.56 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} e x^{3} + 7 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4} +{\left (7 \, c^{3} d^{4} + 8 \, a c^{2} d^{2} e^{2}\right )} x^{2} +{\left (14 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{35 \,{\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*d^3*e*x^3 + 7*a^2*c*d^2*e^2 - 2*a^3*e^4 + (7*c^3*d^4 + 8*a*c^2*d^2*e^2)*x^2 + (14*a*c^2*d^3*e + a^
2*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^2*d^2*e*x + c^2*d^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

Timed out