3.740 \(\int \frac{\sqrt{a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{9/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(7/2)) + (8*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (16*c
^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(3/2))

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Rubi [A]  time = 0.219924, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(9/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(7/2)) + (8*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (16*c
^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(3/2))

Rule 872

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
] - Dist[(c*e*(m - n - 2))/((n + 1)*(c*e*f + c*d*g - b*e*g)), Int[(d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c*x
^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, 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] && LtQ[n, -1] && IntegerQ[2*p]

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{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{9/2}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac{(4 c d) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{7/2}} \, dx}{7 (c d f-a e g)}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac{8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac{\left (8 c^2 d^2\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} (f+g x)^{5/2}} \, dx}{35 (c d f-a e g)^2}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac{8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac{16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.105384, size = 105, normalized size = 0.53 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (15 a^2 e^2 g^2-6 a c d e g (7 f+2 g x)+c^2 d^2 \left (35 f^2+28 f g x+8 g^2 x^2\right )\right )}{105 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(9/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(15*a^2*e^2*g^2 - 6*a*c*d*e*g*(7*f + 2*g*x) + c^2*d^2*(35*f^2 + 28*f*g*x +
8*g^2*x^2)))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(7/2))

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Maple [A]  time = 0.053, size = 169, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{c}^{2}{d}^{2}{g}^{2}{x}^{2}-12\,acde{g}^{2}x+28\,{c}^{2}{d}^{2}fgx+15\,{a}^{2}{e}^{2}{g}^{2}-42\,acdefg+35\,{c}^{2}{d}^{2}{f}^{2} \right ) }{105\,{a}^{3}{e}^{3}{g}^{3}-315\,{a}^{2}cd{e}^{2}f{g}^{2}+315\,a{c}^{2}{d}^{2}e{f}^{2}g-105\,{c}^{3}{d}^{3}{f}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( gx+f \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ex+d}}}} \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)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x)

[Out]

-2/105*(c*d*x+a*e)*(8*c^2*d^2*g^2*x^2-12*a*c*d*e*g^2*x+28*c^2*d^2*f*g*x+15*a^2*e^2*g^2-42*a*c*d*e*f*g+35*c^2*d
^2*f^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(g*x+f)^(7/2)/(a^3*e^3*g^3-3*a^2*c*d*e^2*f*g^2+3*a*c^2*d^2*e*f
^2*g-c^3*d^3*f^3)/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}{\left (g x + f\right )}^{\frac{9}{2}}}\,{d x} \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)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^(9/2)), x)

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Fricas [B]  time = 2.09092, size = 1490, normalized size = 7.53 \begin{align*} \frac{2 \,{\left (8 \, c^{3} d^{3} g^{2} x^{3} + 35 \, a c^{2} d^{2} e f^{2} - 42 \, a^{2} c d e^{2} f g + 15 \, a^{3} e^{3} g^{2} + 4 \,{\left (7 \, c^{3} d^{3} f g - a c^{2} d^{2} e g^{2}\right )} x^{2} +{\left (35 \, c^{3} d^{3} f^{2} - 14 \, a c^{2} d^{2} e f g + 3 \, a^{2} c d e^{2} g^{2}\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} \sqrt{g x + f}}{105 \,{\left (c^{3} d^{4} f^{7} - 3 \, a c^{2} d^{3} e f^{6} g + 3 \, a^{2} c d^{2} e^{2} f^{5} g^{2} - a^{3} d e^{3} f^{4} g^{3} +{\left (c^{3} d^{3} e f^{3} g^{4} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{5} + 3 \, a^{2} c d e^{3} f g^{6} - a^{3} e^{4} g^{7}\right )} x^{5} +{\left (4 \, c^{3} d^{3} e f^{4} g^{3} - a^{3} d e^{3} g^{7} +{\left (c^{3} d^{4} - 12 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{4} - 3 \,{\left (a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} f^{2} g^{5} +{\left (3 \, a^{2} c d^{2} e^{2} - 4 \, a^{3} e^{4}\right )} f g^{6}\right )} x^{4} + 2 \,{\left (3 \, c^{3} d^{3} e f^{5} g^{2} - 2 \, a^{3} d e^{3} f g^{6} +{\left (2 \, c^{3} d^{4} - 9 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{3} - 3 \,{\left (2 \, a c^{2} d^{3} e - 3 \, a^{2} c d e^{3}\right )} f^{3} g^{4} + 3 \,{\left (2 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{2} g^{5}\right )} x^{3} + 2 \,{\left (2 \, c^{3} d^{3} e f^{6} g - 3 \, a^{3} d e^{3} f^{2} g^{5} + 3 \,{\left (c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2}\right )} f^{5} g^{2} - 3 \,{\left (3 \, a c^{2} d^{3} e - 2 \, a^{2} c d e^{3}\right )} f^{4} g^{3} +{\left (9 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4}\right )} f^{3} g^{4}\right )} x^{2} +{\left (c^{3} d^{3} e f^{7} - 4 \, a^{3} d e^{3} f^{3} g^{4} +{\left (4 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{6} g - 3 \,{\left (4 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{5} g^{2} +{\left (12 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{4} g^{3}\right )} x\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)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/105*(8*c^3*d^3*g^2*x^3 + 35*a*c^2*d^2*e*f^2 - 42*a^2*c*d*e^2*f*g + 15*a^3*e^3*g^2 + 4*(7*c^3*d^3*f*g - a*c^2
*d^2*e*g^2)*x^2 + (35*c^3*d^3*f^2 - 14*a*c^2*d^2*e*f*g + 3*a^2*c*d*e^2*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)/(c^3*d^4*f^7 - 3*a*c^2*d^3*e*f^6*g + 3*a^2*c*d^2*e^2*f^5*g^2 - a^3*d*
e^3*f^4*g^3 + (c^3*d^3*e*f^3*g^4 - 3*a*c^2*d^2*e^2*f^2*g^5 + 3*a^2*c*d*e^3*f*g^6 - a^3*e^4*g^7)*x^5 + (4*c^3*d
^3*e*f^4*g^3 - a^3*d*e^3*g^7 + (c^3*d^4 - 12*a*c^2*d^2*e^2)*f^3*g^4 - 3*(a*c^2*d^3*e - 4*a^2*c*d*e^3)*f^2*g^5
+ (3*a^2*c*d^2*e^2 - 4*a^3*e^4)*f*g^6)*x^4 + 2*(3*c^3*d^3*e*f^5*g^2 - 2*a^3*d*e^3*f*g^6 + (2*c^3*d^4 - 9*a*c^2
*d^2*e^2)*f^4*g^3 - 3*(2*a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^3*g^4 + 3*(2*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^5)*x^3 + 2
*(2*c^3*d^3*e*f^6*g - 3*a^3*d*e^3*f^2*g^5 + 3*(c^3*d^4 - 2*a*c^2*d^2*e^2)*f^5*g^2 - 3*(3*a*c^2*d^3*e - 2*a^2*c
*d*e^3)*f^4*g^3 + (9*a^2*c*d^2*e^2 - 2*a^3*e^4)*f^3*g^4)*x^2 + (c^3*d^3*e*f^7 - 4*a^3*d*e^3*f^3*g^4 + (4*c^3*d
^4 - 3*a*c^2*d^2*e^2)*f^6*g - 3*(4*a*c^2*d^3*e - a^2*c*d*e^3)*f^5*g^2 + (12*a^2*c*d^2*e^2 - a^3*e^4)*f^4*g^3)*
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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**(9/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}{\left (g x + f\right )}^{\frac{9}{2}}}\,{d x} \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)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^(9/2)), x)