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

Optimal. Leaf size=267 \[ \frac{32 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}+\frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{12 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 \sqrt{d+e x} (f+g x)^{7/2} (c d f-a e g)} \]

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^(7/2)) + (12*c*d*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(5/2)) + (16*c^2*d^2*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (32*c^3*d^
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*Sqrt[f + g*x])

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Rubi [A]  time = 0.313017, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac{32 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}+\frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{12 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 \sqrt{d+e x} (f+g x)^{7/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^(7/2)) + (12*c*d*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(5/2)) + (16*c^2*d^2*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (32*c^3*d^
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*Sqrt[f + g*x])

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{d+e x}}{(f+g x)^{9/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt{d+e x} (f+g x)^{7/2}}+\frac{(6 c d) \int \frac{\sqrt{d+e x}}{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 (c d f-a e g)}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt{d+e x} (f+g x)^{7/2}}+\frac{12 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^{5/2}}+\frac{\left (24 c^2 d^2\right ) \int \frac{\sqrt{d+e x}}{(f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 (c d f-a e g)^2}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt{d+e x} (f+g x)^{7/2}}+\frac{12 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^{5/2}}+\frac{16 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt{d+e x} (f+g x)^{3/2}}+\frac{\left (16 c^3 d^3\right ) \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}{35 (c d f-a e g)^3}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt{d+e x} (f+g x)^{7/2}}+\frac{12 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^{5/2}}+\frac{16 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt{d+e x} (f+g x)^{3/2}}+\frac{32 c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^4 \sqrt{d+e x} \sqrt{f+g x}}\\ \end{align*}

Mathematica [A]  time = 0.129423, size = 152, normalized size = 0.57 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (3 a^2 c d e^2 g^2 (7 f+2 g x)-5 a^3 e^3 g^3-a c^2 d^2 e g \left (35 f^2+28 f g x+8 g^2 x^2\right )+c^3 d^3 \left (70 f^2 g x+35 f^3+56 f g^2 x^2+16 g^3 x^3\right )\right )}{35 \sqrt{d+e x} (f+g x)^{7/2} (c d f-a e g)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-5*a^3*e^3*g^3 + 3*a^2*c*d*e^2*g^2*(7*f + 2*g*x) - a*c^2*d^2*e*g*(35*f^2 + 2
8*f*g*x + 8*g^2*x^2) + c^3*d^3*(35*f^3 + 70*f^2*g*x + 56*f*g^2*x^2 + 16*g^3*x^3)))/(35*(c*d*f - a*e*g)^4*Sqrt[
d + e*x]*(f + g*x)^(7/2))

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Maple [A]  time = 0.057, size = 260, normalized size = 1. \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{c}^{3}{d}^{3}{g}^{3}{x}^{3}+8\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-56\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-6\,{a}^{2}cd{e}^{2}{g}^{3}x+28\,a{c}^{2}{d}^{2}ef{g}^{2}x-70\,{c}^{3}{d}^{3}{f}^{2}gx+5\,{a}^{3}{e}^{3}{g}^{3}-21\,{a}^{2}cd{e}^{2}f{g}^{2}+35\,a{c}^{2}{d}^{2}e{f}^{2}g-35\,{c}^{3}{d}^{3}{f}^{3} \right ) }{35\,{g}^{4}{e}^{4}{a}^{4}-140\,cd{g}^{3}f{e}^{3}{a}^{3}+210\,{c}^{2}{d}^{2}{g}^{2}{f}^{2}{e}^{2}{a}^{2}-140\,{c}^{3}{d}^{3}g{f}^{3}ea+35\,{c}^{4}{d}^{4}{f}^{4}}\sqrt{ex+d} \left ( gx+f \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(g*x+f)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-2/35*(c*d*x+a*e)*(-16*c^3*d^3*g^3*x^3+8*a*c^2*d^2*e*g^3*x^2-56*c^3*d^3*f*g^2*x^2-6*a^2*c*d*e^2*g^3*x+28*a*c^2
*d^2*e*f*g^2*x-70*c^3*d^3*f^2*g*x+5*a^3*e^3*g^3-21*a^2*c*d*e^2*f*g^2+35*a*c^2*d^2*e*f^2*g-35*c^3*d^3*f^3)*(e*x
+d)^(1/2)/(g*x+f)^(7/2)/(a^4*e^4*g^4-4*a^3*c*d*e^3*f*g^3+6*a^2*c^2*d^2*e^2*f^2*g^2-4*a*c^3*d^3*e*f^3*g+c^4*d^4
*f^4)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(g*x+f)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.84995, size = 1871, normalized size = 7.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(g*x+f)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/35*(16*c^3*d^3*g^3*x^3 + 35*c^3*d^3*f^3 - 35*a*c^2*d^2*e*f^2*g + 21*a^2*c*d*e^2*f*g^2 - 5*a^3*e^3*g^3 + 8*(7
*c^3*d^3*f*g^2 - a*c^2*d^2*e*g^3)*x^2 + 2*(35*c^3*d^3*f^2*g - 14*a*c^2*d^2*e*f*g^2 + 3*a^2*c*d*e^2*g^3)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)/(c^4*d^5*f^8 - 4*a*c^3*d^4*e*f^7*g + 6*a^
2*c^2*d^3*e^2*f^6*g^2 - 4*a^3*c*d^2*e^3*f^5*g^3 + a^4*d*e^4*f^4*g^4 + (c^4*d^4*e*f^4*g^4 - 4*a*c^3*d^3*e^2*f^3
*g^5 + 6*a^2*c^2*d^2*e^3*f^2*g^6 - 4*a^3*c*d*e^4*f*g^7 + a^4*e^5*g^8)*x^5 + (4*c^4*d^4*e*f^5*g^3 + a^4*d*e^4*g
^8 + (c^4*d^5 - 16*a*c^3*d^3*e^2)*f^4*g^4 - 4*(a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*f^3*g^5 + 2*(3*a^2*c^2*d^3*e^2
- 8*a^3*c*d*e^4)*f^2*g^6 - 4*(a^3*c*d^2*e^3 - a^4*e^5)*f*g^7)*x^4 + 2*(3*c^4*d^4*e*f^6*g^2 + 2*a^4*d*e^4*f*g^
7 + 2*(c^4*d^5 - 6*a*c^3*d^3*e^2)*f^5*g^3 - 2*(4*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^4*g^4 + 12*(a^2*c^2*d^3*e^
2 - a^3*c*d*e^4)*f^3*g^5 - (8*a^3*c*d^2*e^3 - 3*a^4*e^5)*f^2*g^6)*x^3 + 2*(2*c^4*d^4*e*f^7*g + 3*a^4*d*e^4*f^2
*g^6 + (3*c^4*d^5 - 8*a*c^3*d^3*e^2)*f^6*g^2 - 12*(a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^5*g^3 + 2*(9*a^2*c^2*d^3*e
^2 - 4*a^3*c*d*e^4)*f^4*g^4 - 2*(6*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^5)*x^2 + (c^4*d^4*e*f^8 + 4*a^4*d*e^4*f^3*g^
5 + 4*(c^4*d^5 - a*c^3*d^3*e^2)*f^7*g - 2*(8*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^6*g^2 + 4*(6*a^2*c^2*d^3*e^2 -
a^3*c*d*e^4)*f^5*g^3 - (16*a^3*c*d^2*e^3 - a^4*e^5)*f^4*g^4)*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)**(1/2)/(g*x+f)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(g*x+f)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

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