\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx\) [694]

Optimal result

Integrand size = 46, antiderivative size = 179 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=-\frac {2 (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^2 \sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}+\frac {2 (c d f-a e g)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{g^{5/2}} \]

[Out]

2/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g/(e*x+d)^(3/2)+2*(-a*e*g+c*d*f)^(3/2)*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^(5/2)-2*(-a*e*g+c*d*f)*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)/g^2/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {878, 888, 211} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\frac {2 (c d f-a e g)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{5/2}}-\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{g^2 \sqrt {d+e x}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}} \]

[In]

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

[Out]

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

Rule 211

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

Rule 878

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

Rule 888

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {\left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\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)} \, dx}{e^2 g} \\ & = -\frac {2 (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^2 \sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}+\frac {(c d f-a e g)^2 \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}{g^2} \\ & = -\frac {2 (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^2 \sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}+\frac {\left (2 e^2 (c d f-a e g)^2\right ) \text {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 )}{g^2} \\ & = -\frac {2 (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^2 \sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}+\frac {2 (c d f-a e g)^{3/2} \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 )}{g^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {a e+c d x} (4 a e g+c d (-3 f+g x))+3 (c d f-a e g)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{3 g^{5/2} \sqrt {(a e+c d x) (d+e x)}} \]

[In]

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

[Out]

(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[g]*Sqrt[a*e + c*d*x]*(4*a*e*g + c*d*(-3*f + g*x)) + 3*(c*d*f - a*e*g)
^(3/2)*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]]))/(3*g^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.41

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a^{2} e^{2} g^{2}-6 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d e f g +3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f^{2}-\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c d g x -4 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a e g +3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c d f \right )}{3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, g^{2} \sqrt {\left (a e g -c d f \right ) g}}\)	\(253\)

[In]

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

[Out]

-2/3*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a^2*e^2*g^2-6*arctanh
(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c*d*e*f*g+3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2
))*c^2*d^2*f^2-(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c*d*g*x-4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a
*e*g+3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c*d*f)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/g^2/((a*e*g-c*d*f)*g)^
(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.28 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\left [-\frac {3 \, {\left (c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x\right )} \sqrt {-\frac {c d f - a e g}{g}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} g \sqrt {-\frac {c d f - a e g}{g}} - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x - 3 \, c d f + 4 \, a e g\right )} \sqrt {e x + d}}{3 \, {\left (e g^{2} x + d g^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x\right )} \sqrt {\frac {c d f - a e g}{g}} \arctan \left (\frac {\sqrt {e x + d} \sqrt {\frac {c d f - a e g}{g}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\right ) - \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x - 3 \, c d f + 4 \, a e g\right )} \sqrt {e x + d}\right )}}{3 \, {\left (e g^{2} x + d g^{2}\right )}}\right ] \]

[In]

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

[Out]

[-1/3*(3*(c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)*x)*sqrt(-(c*d*f - a*e*g)/g)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*
a*d*e*g - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*g*sqrt(-(c*d*f - a*e*g)/g) - (c*d*e*f -
(c*d^2 + 2*a*e^2)*g)*x)/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*
g*x - 3*c*d*f + 4*a*e*g)*sqrt(e*x + d))/(e*g^2*x + d*g^2), -2/3*(3*(c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)*x)
*sqrt((c*d*f - a*e*g)/g)*arctan(sqrt(e*x + d)*sqrt((c*d*f - a*e*g)/g)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)
*x)) - sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*g*x - 3*c*d*f + 4*a*e*g)*sqrt(e*x + d))/(e*g^2*x + d*g
^2)]

Sympy [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (157) = 314\).

Time = 0.47 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\frac {2 \, {\left (c^{2} d^{2} f^{2} {\left | e \right |} - 2 \, a c d e f g {\left | e \right |} + a^{2} e^{2} g^{2} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{\sqrt {c d f g - a e g^{2}} e g^{2}} - \frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} f^{2} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 6 \, a c d e^{3} f g {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + 3 \, a^{2} e^{4} g^{2} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 3 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c d e f {\left | e \right |} - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c d^{2} g {\left | e \right |} + 4 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a e^{2} g {\left | e \right |}\right )}}{3 \, \sqrt {c d f g - a e g^{2}} e^{3} g^{2}} - \frac {2 \, {\left (3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c d e^{10} f g {\left | e \right |} - 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a e^{11} g^{2} {\left | e \right |} - {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{8} g^{2} {\left | e \right |}\right )}}{3 \, e^{12} g^{3}} \]

[In]

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

[Out]

2*(c^2*d^2*f^2*abs(e) - 2*a*c*d*e*f*g*abs(e) + a^2*e^2*g^2*abs(e))*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e
^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e))/(sqrt(c*d*f*g - a*e*g^2)*e*g^2) - 2/3*(3*c^2*d^2*e^2*f^2*abs(e)*arctan(sqrt
(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 6*a*c*d*e^3*f*g*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sq
rt(c*d*f*g - a*e*g^2)*e)) + 3*a^2*e^4*g^2*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e))
- 3*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c*d*e*f*abs(e) - sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*
g^2)*c*d^2*g*abs(e) + 4*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*e^2*g*abs(e))/(sqrt(c*d*f*g - a*e*g^2
)*e^3*g^2) - 2/3*(3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*e^10*f*g*abs(e) - 3*sqrt((e*x + d)*c*d*e - c*d
^2*e + a*e^3)*a*e^11*g^2*abs(e) - ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*e^8*g^2*abs(e))/(e^12*g^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{\left (f+g\,x\right )\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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