3.724 \(\int \frac{(d+e x)^{3/2}}{(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{4 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{\sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

[Out]

(-2*Sqrt[d + e*x])/((c*d*f - a*e*g)*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (4*g*Sqrt[a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/((c*d*f - a*e*g)^2*Sqrt[d + e*x]*Sqrt[f + g*x])

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Rubi [A]  time = 0.154125, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {868, 860} \[ -\frac{4 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{\sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d + e*x])/((c*d*f - a*e*g)*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (4*g*Sqrt[a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/((c*d*f - a*e*g)^2*Sqrt[d + e*x]*Sqrt[f + g*x])

Rule 868

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

Rule 860

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

Rubi steps

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

Mathematica [A]  time = 0.0523168, size = 64, normalized size = 0.52 \[ -\frac{2 \sqrt{d+e x} (a e g+c d (f+2 g x))}{\sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 97, normalized size = 0.8 \begin{align*} -2\,{\frac{ \left ( 2\,xcdg+aeg+cdf \right ) \left ( cdx+ae \right ) \left ( ex+d \right ) ^{3/2}}{\sqrt{gx+f} \left ({a}^{2}{e}^{2}{g}^{2}-2\,acdefg+{c}^{2}{d}^{2}{f}^{2} \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^(3/2)), x)

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Fricas [B]  time = 1.75026, size = 648, normalized size = 5.23 \begin{align*} -\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d g x + c d f + a e g\right )} \sqrt{e x + d} \sqrt{g x + f}}{a c^{2} d^{3} e f^{3} - 2 \, a^{2} c d^{2} e^{2} f^{2} g + a^{3} d e^{3} f g^{2} +{\left (c^{3} d^{3} e f^{2} g - 2 \, a c^{2} d^{2} e^{2} f g^{2} + a^{2} c d e^{3} g^{3}\right )} x^{3} +{\left (c^{3} d^{3} e f^{3} +{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{2} g -{\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g^{2} +{\left (a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{3}\right )} x^{2} +{\left (a^{3} d e^{3} g^{3} +{\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} f^{3} -{\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} f^{2} g -{\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f g^{2}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*g*x + c*d*f + a*e*g)*sqrt(e*x + d)*sqrt(g*x + f)/(a*c^2*
d^3*e*f^3 - 2*a^2*c*d^2*e^2*f^2*g + a^3*d*e^3*f*g^2 + (c^3*d^3*e*f^2*g - 2*a*c^2*d^2*e^2*f*g^2 + a^2*c*d*e^3*g
^3)*x^3 + (c^3*d^3*e*f^3 + (c^3*d^4 - a*c^2*d^2*e^2)*f^2*g - (2*a*c^2*d^3*e + a^2*c*d*e^3)*f*g^2 + (a^2*c*d^2*
e^2 + a^3*e^4)*g^3)*x^2 + (a^3*d*e^3*g^3 + (c^3*d^4 + a*c^2*d^2*e^2)*f^3 - (a*c^2*d^3*e + 2*a^2*c*d*e^3)*f^2*g
- (a^2*c*d^2*e^2 - a^3*e^4)*f*g^2)*x)

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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((e*x+d)**(3/2)/(g*x+f)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^(3/2)), x)