3.791 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=351 \[ -\frac {c^2 d^2 \left (6 a e^2 g-c d (5 d g+e f)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{8 g^{3/2} (c d f-a e g)^{7/2}}-\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{8 g \sqrt {d+e x} (f+g x) (c d f-a e g)^3}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{12 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)} \]

[Out]

-1/8*c^2*d^2*(6*a*e^2*g-c*d*(5*d*g+e*f))*arctan(g^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)
^(1/2)/(e*x+d)^(1/2))/g^(3/2)/(-a*e*g+c*d*f)^(7/2)-1/3*(-d*g+e*f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(-
a*e*g+c*d*f)/(g*x+f)^3/(e*x+d)^(1/2)-1/12*(6*a*e^2*g-c*d*(5*d*g+e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/
g/(-a*e*g+c*d*f)^2/(g*x+f)^2/(e*x+d)^(1/2)-1/8*c*d*(6*a*e^2*g-c*d*(5*d*g+e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2)/g/(-a*e*g+c*d*f)^3/(g*x+f)/(e*x+d)^(1/2)

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Rubi [A]  time = 0.56, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {878, 872, 874, 205} \[ -\frac {c^2 d^2 \left (6 a e^2 g-c d (5 d g+e f)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{8 g^{3/2} (c d f-a e g)^{7/2}}-\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{8 g \sqrt {d+e x} (f+g x) (c d f-a e g)^3}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{12 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

-((e*f - d*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) - (
(6*a*e^2*g - c*d*(e*f + 5*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*g*(c*d*f - a*e*g)^2*Sqrt[d +
e*x]*(f + g*x)^2) - (c*d*(6*a*e^2*g - c*d*(e*f + 5*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*g*(c*
d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) - (c^2*d^2*(6*a*e^2*g - c*d*(e*f + 5*d*g))*ArcTan[(Sqrt[g]*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(8*g^(3/2)*(c*d*f - a*e*g)^(7/2))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 874

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{a, b, c, d, e, f, g}, 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]

Rule 878

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(e^2*(e*f - d*g)*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/(g*(n + 1)*(c*e*f + c*d*g
- b*e*g)), x] - Dist[(e*(b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3)))/(g*(n + 1)*(c*e*f + c*d*g - b*
e*g)), Int[(d + e*x)^(m - 1)*(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 - 1, 0] && LtQ[n, -1] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {\left (e \left (\frac {1}{2} c d e^2 f+\frac {11}{2} c d^2 e g-3 e \left (c d^2+a e^2\right ) g\right )\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 g \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )}\\ &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {\left (c d \left (6 a e^2 g-c d (e f+5 d g)\right )\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g (c d f-a e g)^2}\\ &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (c^2 d^2 \left (6 a e^2 g-c d (e f+5 d g)\right )\right ) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 g (c d f-a e g)^3}\\ &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (c^2 d^2 e^2 \left (6 a e^2 g-c d (e f+5 d g)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{8 g (c d f-a e g)^3}\\ &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c^2 d^2 \left (6 a e^2 g-c d (e f+5 d g)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{8 g^{3/2} (c d f-a e g)^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 132, normalized size = 0.38 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {e f-d g}{(f+g x)^3}-\frac {c^2 d^2 \left (c d (5 d g+e f)-6 a e^2 g\right ) \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{(c d f-a e g)^3}\right )}{3 g \sqrt {d+e x} (a e g-c d f)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((e*f - d*g)/(f + g*x)^3 - (c^2*d^2*(-6*a*e^2*g + c*d*(e*f + 5*d*g))*Hypergeome
tric2F1[1/2, 3, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(c*d*f - a*e*g)^3))/(3*g*(-(c*d*f) + a*e*g)*Sqrt[d
+ e*x])

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fricas [B]  time = 1.08, size = 2736, normalized size = 7.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/48*(3*(c^3*d^4*e*f^4 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f^3*g + (c^3*d^3*e^2*f*g^3 + (5*c^3*d^4*e - 6*a*c^2*d
^2*e^3)*g^4)*x^4 + (3*c^3*d^3*e^2*f^2*g^2 + 2*(8*c^3*d^4*e - 9*a*c^2*d^2*e^3)*f*g^3 + (5*c^3*d^5 - 6*a*c^2*d^3
*e^2)*g^4)*x^3 + 3*(c^3*d^3*e^2*f^3*g + 6*(c^3*d^4*e - a*c^2*d^2*e^3)*f^2*g^2 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*
f*g^3)*x^2 + (c^3*d^3*e^2*f^4 + 2*(4*c^3*d^4*e - 3*a*c^2*d^2*e^3)*f^3*g + 3*(5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f^2*
g^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x -
2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f +
d*g)*x)) + 2*(3*c^3*d^3*e*f^4*g + 8*a^3*d*e^3*g^5 - (33*c^3*d^4 - 13*a*c^2*d^2*e^2)*f^3*g^2 + (59*a*c^2*d^3*e
- 20*a^2*c*d*e^3)*f^2*g^3 - 2*(17*a^2*c*d^2*e^2 - 2*a^3*e^4)*f*g^4 - 3*(c^3*d^3*e*f^2*g^3 + (5*c^3*d^4 - 7*a*
c^2*d^2*e^2)*f*g^4 - (5*a*c^2*d^3*e - 6*a^2*c*d*e^3)*g^5)*x^2 - 2*(4*c^3*d^3*e*f^3*g^2 + (20*c^3*d^4 - 29*a*c^
2*d^2*e^2)*f^2*g^3 - (25*a*c^2*d^3*e - 31*a^2*c*d*e^3)*f*g^4 + (5*a^2*c*d^2*e^2 - 6*a^3*e^4)*g^5)*x)*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^7*g^2 - 4*a*c^3*d^4*e*f^6*g^3 + 6*a^2*c^2*d^3*e^2
*f^5*g^4 - 4*a^3*c*d^2*e^3*f^4*g^5 + a^4*d*e^4*f^3*g^6 + (c^4*d^4*e*f^4*g^5 - 4*a*c^3*d^3*e^2*f^3*g^6 + 6*a^2*
c^2*d^2*e^3*f^2*g^7 - 4*a^3*c*d*e^4*f*g^8 + a^4*e^5*g^9)*x^4 + (3*c^4*d^4*e*f^5*g^4 + a^4*d*e^4*g^9 + (c^4*d^5
- 12*a*c^3*d^3*e^2)*f^4*g^5 - 2*(2*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^3*g^6 + 6*(a^2*c^2*d^3*e^2 - 2*a^3*c*d*
e^4)*f^2*g^7 - (4*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^8)*x^3 + 3*(c^4*d^4*e*f^6*g^3 + a^4*d*e^4*f*g^8 + (c^4*d^5 -
4*a*c^3*d^3*e^2)*f^5*g^4 - 2*(2*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^4*g^5 + 2*(3*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^
4)*f^3*g^6 - (4*a^3*c*d^2*e^3 - a^4*e^5)*f^2*g^7)*x^2 + (c^4*d^4*e*f^7*g^2 + 3*a^4*d*e^4*f^2*g^7 + (3*c^4*d^5
- 4*a*c^3*d^3*e^2)*f^6*g^3 - 6*(2*a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^5*g^4 + 2*(9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^
4)*f^4*g^5 - (12*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^6)*x), -1/24*(3*(c^3*d^4*e*f^4 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)
*f^3*g + (c^3*d^3*e^2*f*g^3 + (5*c^3*d^4*e - 6*a*c^2*d^2*e^3)*g^4)*x^4 + (3*c^3*d^3*e^2*f^2*g^2 + 2*(8*c^3*d^4
*e - 9*a*c^2*d^2*e^3)*f*g^3 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*g^4)*x^3 + 3*(c^3*d^3*e^2*f^3*g + 6*(c^3*d^4*e - a
*c^2*d^2*e^3)*f^2*g^2 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f*g^3)*x^2 + (c^3*d^3*e^2*f^4 + 2*(4*c^3*d^4*e - 3*a*c^2
*d^2*e^3)*f^3*g + 3*(5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f^2*g^2)*x)*sqrt(c*d*f*g - a*e*g^2)*arctan(sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*f*g - a*e*g^2)*sqrt(e*x + d)/(c*d*e*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*g*x)
) + (3*c^3*d^3*e*f^4*g + 8*a^3*d*e^3*g^5 - (33*c^3*d^4 - 13*a*c^2*d^2*e^2)*f^3*g^2 + (59*a*c^2*d^3*e - 20*a^2*
c*d*e^3)*f^2*g^3 - 2*(17*a^2*c*d^2*e^2 - 2*a^3*e^4)*f*g^4 - 3*(c^3*d^3*e*f^2*g^3 + (5*c^3*d^4 - 7*a*c^2*d^2*e^
2)*f*g^4 - (5*a*c^2*d^3*e - 6*a^2*c*d*e^3)*g^5)*x^2 - 2*(4*c^3*d^3*e*f^3*g^2 + (20*c^3*d^4 - 29*a*c^2*d^2*e^2)
*f^2*g^3 - (25*a*c^2*d^3*e - 31*a^2*c*d*e^3)*f*g^4 + (5*a^2*c*d^2*e^2 - 6*a^3*e^4)*g^5)*x)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^7*g^2 - 4*a*c^3*d^4*e*f^6*g^3 + 6*a^2*c^2*d^3*e^2*f^5*g^4 -
4*a^3*c*d^2*e^3*f^4*g^5 + a^4*d*e^4*f^3*g^6 + (c^4*d^4*e*f^4*g^5 - 4*a*c^3*d^3*e^2*f^3*g^6 + 6*a^2*c^2*d^2*e^
3*f^2*g^7 - 4*a^3*c*d*e^4*f*g^8 + a^4*e^5*g^9)*x^4 + (3*c^4*d^4*e*f^5*g^4 + a^4*d*e^4*g^9 + (c^4*d^5 - 12*a*c^
3*d^3*e^2)*f^4*g^5 - 2*(2*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^3*g^6 + 6*(a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^2*g
^7 - (4*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^8)*x^3 + 3*(c^4*d^4*e*f^6*g^3 + a^4*d*e^4*f*g^8 + (c^4*d^5 - 4*a*c^3*d^
3*e^2)*f^5*g^4 - 2*(2*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^4*g^5 + 2*(3*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^3*g^6
- (4*a^3*c*d^2*e^3 - a^4*e^5)*f^2*g^7)*x^2 + (c^4*d^4*e*f^7*g^2 + 3*a^4*d*e^4*f^2*g^7 + (3*c^4*d^5 - 4*a*c^3*
d^3*e^2)*f^6*g^3 - 6*(2*a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^5*g^4 + 2*(9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^4*g^5
- (12*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^6)*x)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.05, size = 1142, normalized size = 3.25 \[ -\frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (18 a \,c^{2} d^{2} e^{2} g^{4} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-15 c^{3} d^{4} g^{4} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-3 c^{3} d^{3} e f \,g^{3} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+54 a \,c^{2} d^{2} e^{2} f \,g^{3} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-45 c^{3} d^{4} f \,g^{3} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-9 c^{3} d^{3} e \,f^{2} g^{2} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+54 a \,c^{2} d^{2} e^{2} f^{2} g^{2} x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-45 c^{3} d^{4} f^{2} g^{2} x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-9 c^{3} d^{3} e \,f^{3} g x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+18 a \,c^{2} d^{2} e^{2} f^{3} g \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-15 c^{3} d^{4} f^{3} g \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-3 c^{3} d^{3} e \,f^{4} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-18 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d \,e^{2} g^{3} x^{2}+15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{3} g^{3} x^{2}+3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} e f \,g^{2} x^{2}+12 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{3} g^{3} x -10 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c \,d^{2} e \,g^{3} x -50 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d \,e^{2} f \,g^{2} x +40 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{3} f \,g^{2} x +8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} e \,f^{2} g x +8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} d \,e^{2} g^{3}+4 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{3} f \,g^{2}-26 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c \,d^{2} e f \,g^{2}-16 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d \,e^{2} f^{2} g +33 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{3} f^{2} g -3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} e \,f^{3}\right )}{24 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{3} \left (a e g -c d f \right ) \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {c d x +a e}\, g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(g*x+f)^4/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2),x)

[Out]

-1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(-3*c^2*d^2*e*f^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-15*a
rctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)*g)*x^3*c^3*d^4*g^4-15*arctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f
)*g)^(1/2)*g)*c^3*d^4*f^3*g-3*arctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)*g)*c^3*d^3*e*f^4-50*x*a*c*d*e^
2*f*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-45*arctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)*g)*x^2*
c^3*d^4*f*g^3-45*arctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)*g)*x*c^3*d^4*f^2*g^2+15*x^2*c^2*d^3*g^3*(c*
d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+12*x*a^2*e^3*g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+8*a^2*d*e^2*
g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+4*a^2*e^3*f*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+33*c^2
*d^3*f^2*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-18*x^2*a*c*d*e^2*g^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^
(1/2)+40*x*c^2*d^3*f*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+18*arctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)
*g)^(1/2)*g)*x^3*a*c^2*d^2*e^2*g^4-3*arctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)*g)*x^3*c^3*d^3*e*f*g^3-
9*arctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)*g)*x^2*c^3*d^3*e*f^2*g^2-9*arctanh((c*d*x+a*e)^(1/2)/((a*e
*g-c*d*f)*g)^(1/2)*g)*x*c^3*d^3*e*f^3*g+18*arctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)*g)*a*c^2*d^2*e^2*
f^3*g+3*x^2*c^2*d^2*e*f*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-10*x*a*c*d^2*e*g^3*(c*d*x+a*e)^(1/2)*((a
*e*g-c*d*f)*g)^(1/2)+8*x*c^2*d^2*e*f^2*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-26*a*c*d^2*e*f*g^2*(c*d*x+a
*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-16*a*c*d*e^2*f^2*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+54*arctanh((c*d
*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)*g)*x^2*a*c^2*d^2*e^2*f*g^3+54*arctanh((c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g
)^(1/2)*g)*x*a*c^2*d^2*e^2*f^2*g^2)/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^3/g/(a*e*g-c*d*f)/(a^2*e^2*g
^2-2*a*c*d*e*f*g+c^2*d^2*f^2)/(c*d*x+a*e)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/((f + g*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)

[Out]

int((d + e*x)^(3/2)/((f + g*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(g*x+f)**4/(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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