3.726 \(\int \frac{(d+e x)^{3/2}}{(f+g x)^{7/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=262 \[ -\frac{32 c^2 d^2 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}-\frac{16 c d g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}-\frac{12 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

[Out]

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

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Rubi [A]  time = 0.329366, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {868, 872, 860} \[ -\frac{32 c^2 d^2 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}-\frac{16 c d g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}-\frac{12 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(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)*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (12*g*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(5/2)) - (16*c*d*g*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(3/2)) - (32*c^2*d^2*g*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)^4*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 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{(d+e x)^{3/2}}{(f+g x)^{7/2} \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) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{(6 g) \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}{c d f-a e g}\\ &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{12 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^{5/2}}-\frac{(24 c d g) \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}{5 (c d f-a e g)^2}\\ &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{12 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^{5/2}}-\frac{16 c d g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^3 \sqrt{d+e x} (f+g x)^{3/2}}-\frac{\left (16 c^2 d^2 g\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}{5 (c d f-a e g)^3}\\ &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{12 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^{5/2}}-\frac{16 c d g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^3 \sqrt{d+e x} (f+g x)^{3/2}}-\frac{32 c^2 d^2 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^4 \sqrt{d+e x} \sqrt{f+g x}}\\ \end{align*}

Mathematica [A]  time = 0.104442, size = 150, normalized size = 0.57 \[ -\frac{2 \sqrt{d+e x} \left (-a^2 c d e^2 g^2 (5 f+2 g x)+a^3 e^3 g^3+a c^2 d^2 e g \left (15 f^2+20 f g x+8 g^2 x^2\right )+c^3 d^3 \left (30 f^2 g x+5 f^3+40 f g^2 x^2+16 g^3 x^3\right )\right )}{5 (f+g x)^{5/2} \sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(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^3*e^3*g^3 - a^2*c*d*e^2*g^2*(5*f + 2*g*x) + a*c^2*d^2*e*g*(15*f^2 + 20*f*g*x + 8*g^2*x^2)
+ c^3*d^3*(5*f^3 + 30*f^2*g*x + 40*f*g^2*x^2 + 16*g^3*x^3)))/(5*(c*d*f - a*e*g)^4*Sqrt[(a*e + c*d*x)*(d + e*x
)]*(f + g*x)^(5/2))

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Maple [A]  time = 0.055, size = 259, 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}+40\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-2\,{a}^{2}cd{e}^{2}{g}^{3}x+20\,a{c}^{2}{d}^{2}ef{g}^{2}x+30\,{c}^{3}{d}^{3}{f}^{2}gx+{a}^{3}{e}^{3}{g}^{3}-5\,{a}^{2}cd{e}^{2}f{g}^{2}+15\,a{c}^{2}{d}^{2}e{f}^{2}g+5\,{c}^{3}{d}^{3}{f}^{3} \right ) }{5\,{g}^{4}{e}^{4}{a}^{4}-20\,cd{g}^{3}f{e}^{3}{a}^{3}+30\,{c}^{2}{d}^{2}{g}^{2}{f}^{2}{e}^{2}{a}^{2}-20\,{c}^{3}{d}^{3}g{f}^{3}ea+5\,{c}^{4}{d}^{4}{f}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( gx+f \right ) ^{-{\frac{5}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

-2/5*(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+40*c^3*d^3*f*g^2*x^2-2*a^2*c*d*e^2*g^3*x+20*a*c^2*d
^2*e*f*g^2*x+30*c^3*d^3*f^2*g*x+a^3*e^3*g^3-5*a^2*c*d*e^2*f*g^2+15*a*c^2*d^2*e*f^2*g+5*c^3*d^3*f^3)*(e*x+d)^(3
/2)/(g*x+f)^(5/2)/(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)^(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{7}{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)^(7/2)/(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)^(7/2)), x)

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Fricas [B]  time = 2.17315, size = 2063, normalized size = 7.87 \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)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

-2/5*(16*c^3*d^3*g^3*x^3 + 5*c^3*d^3*f^3 + 15*a*c^2*d^2*e*f^2*g - 5*a^2*c*d*e^2*f*g^2 + a^3*e^3*g^3 + 8*(5*c^3
*d^3*f*g^2 + a*c^2*d^2*e*g^3)*x^2 + 2*(15*c^3*d^3*f^2*g + 10*a*c^2*d^2*e*f*g^2 - a^2*c*d*e^2*g^3)*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)/(a*c^4*d^5*e*f^7 - 4*a^2*c^3*d^4*e^2*f^6*g + 6*
a^3*c^2*d^3*e^3*f^5*g^2 - 4*a^4*c*d^2*e^4*f^4*g^3 + a^5*d*e^5*f^3*g^4 + (c^5*d^5*e*f^4*g^3 - 4*a*c^4*d^4*e^2*f
^3*g^4 + 6*a^2*c^3*d^3*e^3*f^2*g^5 - 4*a^3*c^2*d^2*e^4*f*g^6 + a^4*c*d*e^5*g^7)*x^5 + (3*c^5*d^5*e*f^5*g^2 + (
c^5*d^6 - 11*a*c^4*d^4*e^2)*f^4*g^3 - 2*(2*a*c^4*d^5*e - 7*a^2*c^3*d^3*e^3)*f^3*g^4 + 6*(a^2*c^3*d^4*e^2 - a^3
*c^2*d^2*e^4)*f^2*g^5 - (4*a^3*c^2*d^3*e^3 + a^4*c*d*e^5)*f*g^6 + (a^4*c*d^2*e^4 + a^5*e^6)*g^7)*x^4 + (3*c^5*
d^5*e*f^6*g + a^5*d*e^5*g^7 + 3*(c^5*d^6 - 3*a*c^4*d^4*e^2)*f^5*g^2 - (11*a*c^4*d^5*e - 6*a^2*c^3*d^3*e^3)*f^4
*g^3 + 2*(7*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4)*f^3*g^4 - 3*(2*a^3*c^2*d^3*e^3 + 3*a^4*c*d*e^5)*f^2*g^5 - (a^
4*c*d^2*e^4 - 3*a^5*e^6)*f*g^6)*x^3 + (c^5*d^5*e*f^7 + 3*a^5*d*e^5*f*g^6 + (3*c^5*d^6 - a*c^4*d^4*e^2)*f^6*g -
3*(3*a*c^4*d^5*e + 2*a^2*c^3*d^3*e^3)*f^5*g^2 + 2*(3*a^2*c^3*d^4*e^2 + 7*a^3*c^2*d^2*e^4)*f^4*g^3 + (6*a^3*c^
2*d^3*e^3 - 11*a^4*c*d*e^5)*f^3*g^4 - 3*(3*a^4*c*d^2*e^4 - a^5*e^6)*f^2*g^5)*x^2 + (3*a^5*d*e^5*f^2*g^5 + (c^5
*d^6 + a*c^4*d^4*e^2)*f^7 - (a*c^4*d^5*e + 4*a^2*c^3*d^3*e^3)*f^6*g - 6*(a^2*c^3*d^4*e^2 - a^3*c^2*d^2*e^4)*f^
5*g^2 + 2*(7*a^3*c^2*d^3*e^3 - 2*a^4*c*d*e^5)*f^4*g^3 - (11*a^4*c*d^2*e^4 - a^5*e^6)*f^3*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)**(3/2)/(g*x+f)**(7/2)/(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{7}{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)^(7/2)/(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)^(7/2)), x)