3.682 \(\int \frac{(f+g x) \sqrt{a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=125 \[ \frac{2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d e \sqrt{d+e x}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{15 c^2 d^2 e (d+e x)^{3/2}} \]

[Out]

(-2*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(15*c^2*d^2*e*(d + e*x)^(
3/2)) + (2*g*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*c*d*e*Sqrt[d + e*x])

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Rubi [A]  time = 0.0951645, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {794, 648} \[ \frac{2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d e \sqrt{d+e x}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{15 c^2 d^2 e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(15*c^2*d^2*e*(d + e*x)^(
3/2)) + (2*g*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*c*d*e*Sqrt[d + e*x])

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 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{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx &=\frac{2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d e \sqrt{d+e x}}+\frac{1}{5} \left (5 f-\frac{3 d g}{e}-\frac{2 a e g}{c d}\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx\\ &=\frac{2 \left (5 f-\frac{3 d g}{e}-\frac{2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 c d (d+e x)^{3/2}}+\frac{2 g \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 c d e \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0534116, size = 54, normalized size = 0.43 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} (c d (5 f+3 g x)-2 a e g)}{15 c^2 d^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-2*a*e*g + c*d*(5*f + 3*g*x)))/(15*c^2*d^2*(d + e*x)^(3/2))

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Maple [A]  time = 0.047, size = 67, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -3\,xcdg+2\,aeg-5\,cdf \right ) }{15\,{c}^{2}{d}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(c*d*x+a*e)*(-3*c*d*g*x+2*a*e*g-5*c*d*f)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/c^2/d^2/(e*x+d)^(1/2)

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Maxima [A]  time = 1.10425, size = 88, normalized size = 0.7 \begin{align*} \frac{2 \,{\left (c d x + a e\right )}^{\frac{3}{2}} f}{3 \, c d} + \frac{2 \,{\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt{c d x + a e} g}{15 \, c^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(c*d*x + a*e)^(3/2)*f/(c*d) + 2/15*(3*c^2*d^2*x^2 + a*c*d*e*x - 2*a^2*e^2)*sqrt(c*d*x + a*e)*g/(c^2*d^2)

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Fricas [A]  time = 1.61304, size = 221, normalized size = 1.77 \begin{align*} \frac{2 \,{\left (3 \, c^{2} d^{2} g x^{2} + 5 \, a c d e f - 2 \, a^{2} e^{2} g +{\left (5 \, c^{2} d^{2} f + a c d e g\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}}{15 \,{\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((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*c^2*d^2*g*x^2 + 5*a*c*d*e*f - 2*a^2*e^2*g + (5*c^2*d^2*f + a*c*d*e*g)*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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )}{\sqrt{d + e x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)/sqrt(d + e*x), x)

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

[Out]

Timed out