3.8.90 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [790]

Optimal. Leaf size=261 \[ -\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {c d \left (4 a e^2 g-c d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 g^{3/2} (c d f-a e g)^{5/2}} \]

[Out]

-1/4*c*d*(4*a*e^2*g-c*d*(3*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)^(5/2)-1/2*(-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)^2/(e*x+d)^(1/2)-1/4*(4*a*e^2*g-c*d*(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)^2/(g*x+f)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {892, 886, 888, 211} \begin {gather*} -\frac {c d \left (4 a e^2 g-c d (3 d g+e f)\right ) \text {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 )}{4 g^{3/2} (c d f-a e g)^{5/2}}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (3 d g+e f)\right )}{4 g \sqrt {d+e x} (f+g x) (c d f-a e g)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((e*f - d*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^2) -
((4*a*e^2*g - c*d*(e*f + 3*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*g*(c*d*f - a*e*g)^2*Sqrt[d +
e*x]*(f + g*x)) - (c*d*(4*a*e^2*g - c*d*(e*f + 3*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])])/(4*g^(3/2)*(c*d*f - a*e*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 886

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 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]

Rule 892

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)^3 \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}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {\left (e \left (\frac {1}{2} c d e^2 f+\frac {7}{2} c d^2 e g-2 e \left (c d^2+a e^2\right ) 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}{2 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}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {\left (c d \left (4 a e^2 g-c d (e f+3 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}{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}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {\left (c d e^2 \left (4 a e^2 g-c d (e f+3 d g)\right )\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 )}{4 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}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {c d \left (4 a e^2 g-c d (e f+3 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 )}{4 g^{3/2} (c d f-a e g)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.92, size = 200, normalized size = 0.77 \begin {gather*} \frac {c d \sqrt {d+e x} \left (\frac {\sqrt {g} (a e+c d x) (-2 a e g (d g+e (f+2 g x))+c d (e f (-f+g x)+d g (5 f+3 g x)))}{c d (c d f-a e g)^2 (f+g x)^2}+\frac {\left (-4 a e^2 g+c d (e f+3 d g)\right ) \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2}}\right )}{4 g^{3/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(662\) vs. \(2(235)=470\).
time = 0.14, size = 663, normalized size = 2.54

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (4 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d \,e^{2} g^{3} x^{2}-3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{3} g^{3} x^{2}-\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} e f \,g^{2} x^{2}+8 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d \,e^{2} f \,g^{2} x -6 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{3} f \,g^{2} x -2 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} e \,f^{2} g x +4 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d \,e^{2} f^{2} g -3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{3} f^{2} g -\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} e \,f^{3}-4 a \,e^{2} g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+3 c \,d^{2} g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+c d e f g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-2 a d e \,g^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-2 a \,e^{2} f g \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+5 c \,d^{2} f g \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-c d e \,f^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\right )}{4 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{2} g \left (a e g -c d f \right )^{2} \sqrt {c d x +a e}}\)	\(663\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((c*d*x+a*e)*(e*x+d))^(1/2)*(4*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c*d*e^2*g^3*x^2-3*ar
ctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^3*g^3*x^2-arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*
g)^(1/2))*c^2*d^2*e*f*g^2*x^2+8*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c*d*e^2*f*g^2*x-6*arcta
nh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^3*f*g^2*x-2*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g
)^(1/2))*c^2*d^2*e*f^2*g*x+4*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c*d*e^2*f^2*g-3*arctanh(g*
(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^3*f^2*g-arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*
c^2*d^2*e*f^3-4*a*e^2*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+3*c*d^2*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-
c*d*f)*g)^(1/2)+c*d*e*f*g*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-2*a*d*e*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*
d*f)*g)^(1/2)-2*a*e^2*f*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+5*c*d^2*f*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*
f)*g)^(1/2)-c*d*e*f^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2))/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)
^2/g/(a*e*g-c*d*f)^2/(c*d*x+a*e)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (244) = 488\).
time = 2.28, size = 1761, normalized size = 6.75 \begin {gather*} \left [\frac {{\left (3 \, c^{2} d^{4} g^{3} x^{2} + 6 \, c^{2} d^{4} f g^{2} x + 3 \, c^{2} d^{4} f^{2} g - 4 \, {\left (a c d g^{3} x^{3} + 2 \, a c d f g^{2} x^{2} + a c d f^{2} g x\right )} e^{3} + {\left (c^{2} d^{2} f g^{2} x^{3} - 4 \, a c d^{2} f^{2} g + 2 \, {\left (c^{2} d^{2} f^{2} g - 2 \, a c d^{2} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} f^{3} - 8 \, a c d^{2} f g^{2}\right )} x\right )} e^{2} + {\left (3 \, c^{2} d^{3} g^{3} x^{3} + 7 \, c^{2} d^{3} f g^{2} x^{2} + 5 \, c^{2} d^{3} f^{2} g x + c^{2} d^{3} f^{3}\right )} e\right )} \sqrt {-c d f g + a g^{2} e} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e + 2 \, \sqrt {-c d f g + a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) + 2 \, {\left (3 \, c^{2} d^{3} f g^{3} x + 5 \, c^{2} d^{3} f^{2} g^{2} + 2 \, {\left (2 \, a^{2} g^{4} x + a^{2} f g^{3}\right )} e^{3} - {\left (5 \, a c d f g^{3} x + a c d f^{2} g^{2} - 2 \, a^{2} d g^{4}\right )} e^{2} - {\left (c^{2} d^{2} f^{3} g + 7 \, a c d^{2} f g^{3} - {\left (c^{2} d^{2} f^{2} g^{2} - 3 \, a c d^{2} g^{4}\right )} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{8 \, {\left (c^{3} d^{4} f^{3} g^{4} x^{2} + 2 \, c^{3} d^{4} f^{4} g^{3} x + c^{3} d^{4} f^{5} g^{2} - {\left (a^{3} g^{7} x^{3} + 2 \, a^{3} f g^{6} x^{2} + a^{3} f^{2} g^{5} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{6} x^{3} - a^{3} d f^{2} g^{5} + {\left (6 \, a^{2} c d f^{2} g^{5} - a^{3} d g^{7}\right )} x^{2} + {\left (3 \, a^{2} c d f^{3} g^{4} - 2 \, a^{3} d f g^{6}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{5} x^{3} - a^{2} c d^{2} f^{3} g^{4} + {\left (2 \, a c^{2} d^{2} f^{3} g^{4} - a^{2} c d^{2} f g^{6}\right )} x^{2} + {\left (a c^{2} d^{2} f^{4} g^{3} - 2 \, a^{2} c d^{2} f^{2} g^{5}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{4} x^{3} - 3 \, a c^{2} d^{3} f^{4} g^{3} + {\left (2 \, c^{3} d^{3} f^{4} g^{3} - 3 \, a c^{2} d^{3} f^{2} g^{5}\right )} x^{2} + {\left (c^{3} d^{3} f^{5} g^{2} - 6 \, a c^{2} d^{3} f^{3} g^{4}\right )} x\right )} e\right )}}, -\frac {{\left (3 \, c^{2} d^{4} g^{3} x^{2} + 6 \, c^{2} d^{4} f g^{2} x + 3 \, c^{2} d^{4} f^{2} g - 4 \, {\left (a c d g^{3} x^{3} + 2 \, a c d f g^{2} x^{2} + a c d f^{2} g x\right )} e^{3} + {\left (c^{2} d^{2} f g^{2} x^{3} - 4 \, a c d^{2} f^{2} g + 2 \, {\left (c^{2} d^{2} f^{2} g - 2 \, a c d^{2} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} f^{3} - 8 \, a c d^{2} f g^{2}\right )} x\right )} e^{2} + {\left (3 \, c^{2} d^{3} g^{3} x^{3} + 7 \, c^{2} d^{3} f g^{2} x^{2} + 5 \, c^{2} d^{3} f^{2} g x + c^{2} d^{3} f^{3}\right )} e\right )} \sqrt {c d f g - a g^{2} e} \arctan \left (\frac {\sqrt {c d f g - a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) - {\left (3 \, c^{2} d^{3} f g^{3} x + 5 \, c^{2} d^{3} f^{2} g^{2} + 2 \, {\left (2 \, a^{2} g^{4} x + a^{2} f g^{3}\right )} e^{3} - {\left (5 \, a c d f g^{3} x + a c d f^{2} g^{2} - 2 \, a^{2} d g^{4}\right )} e^{2} - {\left (c^{2} d^{2} f^{3} g + 7 \, a c d^{2} f g^{3} - {\left (c^{2} d^{2} f^{2} g^{2} - 3 \, a c d^{2} g^{4}\right )} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{4 \, {\left (c^{3} d^{4} f^{3} g^{4} x^{2} + 2 \, c^{3} d^{4} f^{4} g^{3} x + c^{3} d^{4} f^{5} g^{2} - {\left (a^{3} g^{7} x^{3} + 2 \, a^{3} f g^{6} x^{2} + a^{3} f^{2} g^{5} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{6} x^{3} - a^{3} d f^{2} g^{5} + {\left (6 \, a^{2} c d f^{2} g^{5} - a^{3} d g^{7}\right )} x^{2} + {\left (3 \, a^{2} c d f^{3} g^{4} - 2 \, a^{3} d f g^{6}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{5} x^{3} - a^{2} c d^{2} f^{3} g^{4} + {\left (2 \, a c^{2} d^{2} f^{3} g^{4} - a^{2} c d^{2} f g^{6}\right )} x^{2} + {\left (a c^{2} d^{2} f^{4} g^{3} - 2 \, a^{2} c d^{2} f^{2} g^{5}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{4} x^{3} - 3 \, a c^{2} d^{3} f^{4} g^{3} + {\left (2 \, c^{3} d^{3} f^{4} g^{3} - 3 \, a c^{2} d^{3} f^{2} g^{5}\right )} x^{2} + {\left (c^{3} d^{3} f^{5} g^{2} - 6 \, a c^{2} d^{3} f^{3} g^{4}\right )} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*((3*c^2*d^4*g^3*x^2 + 6*c^2*d^4*f*g^2*x + 3*c^2*d^4*f^2*g - 4*(a*c*d*g^3*x^3 + 2*a*c*d*f*g^2*x^2 + a*c*d*
f^2*g*x)*e^3 + (c^2*d^2*f*g^2*x^3 - 4*a*c*d^2*f^2*g + 2*(c^2*d^2*f^2*g - 2*a*c*d^2*g^3)*x^2 + (c^2*d^2*f^3 - 8
*a*c*d^2*f*g^2)*x)*e^2 + (3*c^2*d^3*g^3*x^3 + 7*c^2*d^3*f*g^2*x^2 + 5*c^2*d^3*f^2*g*x + c^2*d^3*f^3)*e)*sqrt(-
c*d*f*g + a*g^2*e)*log(-(c*d^2*g*x - c*d^2*f + 2*a*g*x*e^2 + (c*d*g*x^2 - c*d*f*x + 2*a*d*g)*e + 2*sqrt(-c*d*f
*g + a*g^2*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(d*g*x + d*f + (g*x^2 + f*x)*e)) + 2*
(3*c^2*d^3*f*g^3*x + 5*c^2*d^3*f^2*g^2 + 2*(2*a^2*g^4*x + a^2*f*g^3)*e^3 - (5*a*c*d*f*g^3*x + a*c*d*f^2*g^2 -
2*a^2*d*g^4)*e^2 - (c^2*d^2*f^3*g + 7*a*c*d^2*f*g^3 - (c^2*d^2*f^2*g^2 - 3*a*c*d^2*g^4)*x)*e)*sqrt(c*d^2*x + a
*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^3*d^4*f^3*g^4*x^2 + 2*c^3*d^4*f^4*g^3*x + c^3*d^4*f^5*g^2 - (a^3
*g^7*x^3 + 2*a^3*f*g^6*x^2 + a^3*f^2*g^5*x)*e^4 + (3*a^2*c*d*f*g^6*x^3 - a^3*d*f^2*g^5 + (6*a^2*c*d*f^2*g^5 -
a^3*d*g^7)*x^2 + (3*a^2*c*d*f^3*g^4 - 2*a^3*d*f*g^6)*x)*e^3 - 3*(a*c^2*d^2*f^2*g^5*x^3 - a^2*c*d^2*f^3*g^4 + (
2*a*c^2*d^2*f^3*g^4 - a^2*c*d^2*f*g^6)*x^2 + (a*c^2*d^2*f^4*g^3 - 2*a^2*c*d^2*f^2*g^5)*x)*e^2 + (c^3*d^3*f^3*g
^4*x^3 - 3*a*c^2*d^3*f^4*g^3 + (2*c^3*d^3*f^4*g^3 - 3*a*c^2*d^3*f^2*g^5)*x^2 + (c^3*d^3*f^5*g^2 - 6*a*c^2*d^3*
f^3*g^4)*x)*e), -1/4*((3*c^2*d^4*g^3*x^2 + 6*c^2*d^4*f*g^2*x + 3*c^2*d^4*f^2*g - 4*(a*c*d*g^3*x^3 + 2*a*c*d*f*
g^2*x^2 + a*c*d*f^2*g*x)*e^3 + (c^2*d^2*f*g^2*x^3 - 4*a*c*d^2*f^2*g + 2*(c^2*d^2*f^2*g - 2*a*c*d^2*g^3)*x^2 +
(c^2*d^2*f^3 - 8*a*c*d^2*f*g^2)*x)*e^2 + (3*c^2*d^3*g^3*x^3 + 7*c^2*d^3*f*g^2*x^2 + 5*c^2*d^3*f^2*g*x + c^2*d^
3*f^3)*e)*sqrt(c*d*f*g - a*g^2*e)*arctan(sqrt(c*d*f*g - a*g^2*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*s
qrt(x*e + d)/(c*d^2*g*x + a*g*x*e^2 + (c*d*g*x^2 + a*d*g)*e)) - (3*c^2*d^3*f*g^3*x + 5*c^2*d^3*f^2*g^2 + 2*(2*
a^2*g^4*x + a^2*f*g^3)*e^3 - (5*a*c*d*f*g^3*x + a*c*d*f^2*g^2 - 2*a^2*d*g^4)*e^2 - (c^2*d^2*f^3*g + 7*a*c*d^2*
f*g^3 - (c^2*d^2*f^2*g^2 - 3*a*c*d^2*g^4)*x)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^
3*d^4*f^3*g^4*x^2 + 2*c^3*d^4*f^4*g^3*x + c^3*d^4*f^5*g^2 - (a^3*g^7*x^3 + 2*a^3*f*g^6*x^2 + a^3*f^2*g^5*x)*e^
4 + (3*a^2*c*d*f*g^6*x^3 - a^3*d*f^2*g^5 + (6*a^2*c*d*f^2*g^5 - a^3*d*g^7)*x^2 + (3*a^2*c*d*f^3*g^4 - 2*a^3*d*
f*g^6)*x)*e^3 - 3*(a*c^2*d^2*f^2*g^5*x^3 - a^2*c*d^2*f^3*g^4 + (2*a*c^2*d^2*f^3*g^4 - a^2*c*d^2*f*g^6)*x^2 + (
a*c^2*d^2*f^4*g^3 - 2*a^2*c*d^2*f^2*g^5)*x)*e^2 + (c^3*d^3*f^3*g^4*x^3 - 3*a*c^2*d^3*f^4*g^3 + (2*c^3*d^3*f^4*
g^3 - 3*a*c^2*d^3*f^2*g^5)*x^2 + (c^3*d^3*f^5*g^2 - 6*a*c^2*d^3*f^3*g^4)*x)*e)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(g*x+f)**3/(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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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