3.8.91 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^4 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [791]

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

[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 = 0.36, 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 = {892, 886, 888, 211} \begin {gather*} -\frac {c^2 d^2 \left (6 a e^2 g-c d (5 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 )}{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)} \end {gather*}

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]

-1/3*((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)^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 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)^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 ) \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 )}{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 [A]
time = 1.19, size = 279, normalized size = 0.79 \begin {gather*} \frac {c^2 d^2 \sqrt {d+e x} \left (\frac {\sqrt {g} (a e+c d x) \left (4 a^2 e^2 g^2 (2 d g+e (f+3 g x))-2 a c d e g \left (d g (13 f+5 g x)+e \left (8 f^2+25 f g x+9 g^2 x^2\right )\right )+c^2 d^2 \left (e f \left (-3 f^2+8 f g x+3 g^2 x^2\right )+d g \left (33 f^2+40 f g x+15 g^2 x^2\right )\right )\right )}{c^2 d^2 (c d f-a e g)^3 (f+g x)^3}+\frac {3 \left (-6 a e^2 g+c d (e f+5 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)^{7/2}}\right )}{24 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)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(c^2*d^2*Sqrt[d + e*x]*((Sqrt[g]*(a*e + c*d*x)*(4*a^2*e^2*g^2*(2*d*g + e*(f + 3*g*x)) - 2*a*c*d*e*g*(d*g*(13*f
+ 5*g*x) + e*(8*f^2 + 25*f*g*x + 9*g^2*x^2)) + c^2*d^2*(e*f*(-3*f^2 + 8*f*g*x + 3*g^2*x^2) + d*g*(33*f^2 + 40
*f*g*x + 15*g^2*x^2))))/(c^2*d^2*(c*d*f - a*e*g)^3*(f + g*x)^3) + (3*(-6*a*e^2*g + c*d*(e*f + 5*d*g))*Sqrt[a*e
+ c*d*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(c*d*f - a*e*g)^(7/2)))/(24*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. \(1131\) vs. \(2(319)=638\).
time = 0.14, size = 1132, normalized size = 3.23

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-45 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{4} f \,g^{3} x^{2}-45 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{4} f^{2} g^{2} x +15 c^{2} d^{3} g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-3 c^{2} d^{2} e \,f^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+8 a^{2} d \,e^{2} g^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+4 a^{2} e^{3} f \,g^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+33 c^{2} d^{3} f^{2} g \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+12 a^{2} e^{3} g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+40 c^{2} d^{3} f \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-50 a c d \,e^{2} f \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+54 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e^{2} f \,g^{3} x^{2}+54 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e^{2} f^{2} g^{2} x -18 a c d \,e^{2} g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+3 c^{2} d^{2} e f \,g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-10 a c \,d^{2} e \,g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+8 c^{2} d^{2} e \,f^{2} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-26 a c \,d^{2} e f \,g^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-16 a c d \,e^{2} f^{2} g \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} e f \,g^{3} x^{3}-9 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} e \,f^{2} g^{2} x^{2}-9 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} e \,f^{3} g x +18 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e^{2} f^{3} g +18 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e^{2} g^{4} x^{3}-15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{4} g^{4} x^{3}-15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{4} f^{3} g -3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} e \,f^{4}\right )}{24 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{3} g \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}}\)	\(1132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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,method=_RETURNVERBOSE)

[Out]

-1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*c^2*d^3*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(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)-45*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^4*f*
g^3*x^2-45*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^4*f^2*g^2*x+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)+12*a^2*e^3*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-3*arctanh(g*(c
*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*e*f*g^3*x^3-9*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(
1/2))*c^3*d^3*e*f^2*g^2*x^2-9*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*e*f^3*g*x+18*arctan
h(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e^2*f^3*g+40*c^2*d^3*f*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*
g-c*d*f)*g)^(1/2)-50*a*c*d*e^2*f*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+18*arctanh(g*(c*d*x+a*e)^(1/2
)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e^2*g^4*x^3+54*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2
*d^2*e^2*f*g^3*x^2+54*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e^2*f^2*g^2*x-18*a*c*d*e^
2*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+3*c^2*d^2*e*f*g^2*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^
(1/2)-10*a*c*d^2*e*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+8*c^2*d^2*e*f^2*g*x*(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)-15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^4*g^4*x^3-15*arct
anh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^4*f^3*g-3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)
^(1/2))*c^3*d^3*e*f^4)/(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 \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)^4/(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)^4), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1414 vs. \(2 (330) = 660\).
time = 3.74, size = 2867, normalized size = 8.17 \begin {gather*} \text {Too large to display} \end {gather*}

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*(5*c^3*d^5*g^4*x^3 + 15*c^3*d^5*f*g^3*x^2 + 15*c^3*d^5*f^2*g^2*x + 5*c^3*d^5*f^3*g - 6*(a*c^2*d^2*g^
4*x^4 + 3*a*c^2*d^2*f*g^3*x^3 + 3*a*c^2*d^2*f^2*g^2*x^2 + a*c^2*d^2*f^3*g*x)*e^3 + (c^3*d^3*f*g^3*x^4 - 6*a*c^
2*d^3*f^3*g + 3*(c^3*d^3*f^2*g^2 - 2*a*c^2*d^3*g^4)*x^3 + 3*(c^3*d^3*f^3*g - 6*a*c^2*d^3*f*g^3)*x^2 + (c^3*d^3
*f^4 - 18*a*c^2*d^3*f^2*g^2)*x)*e^2 + (5*c^3*d^4*g^4*x^4 + 16*c^3*d^4*f*g^3*x^3 + 18*c^3*d^4*f^2*g^2*x^2 + 8*c
^3*d^4*f^3*g*x + c^3*d^4*f^4)*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*(15*c^3*d^4*f*g^4*x^2 + 40*c^3*d^4*f^2*g^3*x + 33*c^3*d^4*f^3*g^2 - 4*(
3*a^3*g^5*x + a^3*f*g^4)*e^4 + 2*(9*a^2*c*d*g^5*x^2 + 31*a^2*c*d*f*g^4*x + 10*a^2*c*d*f^2*g^3 - 4*a^3*d*g^5)*e
^3 - (21*a*c^2*d^2*f*g^4*x^2 + 13*a*c^2*d^2*f^3*g^2 - 34*a^2*c*d^2*f*g^4 + 2*(29*a*c^2*d^2*f^2*g^3 - 5*a^2*c*d
^2*g^5)*x)*e^2 - (3*c^3*d^3*f^4*g + 59*a*c^2*d^3*f^2*g^3 - 3*(c^3*d^3*f^2*g^3 - 5*a*c^2*d^3*g^5)*x^2 - 2*(4*c^
3*d^3*f^3*g^2 - 25*a*c^2*d^3*f*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^4*d^5*
f^4*g^5*x^3 + 3*c^4*d^5*f^5*g^4*x^2 + 3*c^4*d^5*f^6*g^3*x + c^4*d^5*f^7*g^2 + (a^4*g^9*x^4 + 3*a^4*f*g^8*x^3 +
3*a^4*f^2*g^7*x^2 + a^4*f^3*g^6*x)*e^5 - (4*a^3*c*d*f*g^8*x^4 - a^4*d*f^3*g^6 + (12*a^3*c*d*f^2*g^7 - a^4*d*g
^9)*x^3 + 3*(4*a^3*c*d*f^3*g^6 - a^4*d*f*g^8)*x^2 + (4*a^3*c*d*f^4*g^5 - 3*a^4*d*f^2*g^7)*x)*e^4 + 2*(3*a^2*c^
2*d^2*f^2*g^7*x^4 - 2*a^3*c*d^2*f^4*g^5 + (9*a^2*c^2*d^2*f^3*g^6 - 2*a^3*c*d^2*f*g^8)*x^3 + 3*(3*a^2*c^2*d^2*f
^4*g^5 - 2*a^3*c*d^2*f^2*g^7)*x^2 + 3*(a^2*c^2*d^2*f^5*g^4 - 2*a^3*c*d^2*f^3*g^6)*x)*e^3 - 2*(2*a*c^3*d^3*f^3*
g^6*x^4 - 3*a^2*c^2*d^3*f^5*g^4 + 3*(2*a*c^3*d^3*f^4*g^5 - a^2*c^2*d^3*f^2*g^7)*x^3 + 3*(2*a*c^3*d^3*f^5*g^4 -
3*a^2*c^2*d^3*f^3*g^6)*x^2 + (2*a*c^3*d^3*f^6*g^3 - 9*a^2*c^2*d^3*f^4*g^5)*x)*e^2 + (c^4*d^4*f^4*g^5*x^4 - 4*
a*c^3*d^4*f^6*g^3 + (3*c^4*d^4*f^5*g^4 - 4*a*c^3*d^4*f^3*g^6)*x^3 + 3*(c^4*d^4*f^6*g^3 - 4*a*c^3*d^4*f^4*g^5)*
x^2 + (c^4*d^4*f^7*g^2 - 12*a*c^3*d^4*f^5*g^4)*x)*e), -1/24*(3*(5*c^3*d^5*g^4*x^3 + 15*c^3*d^5*f*g^3*x^2 + 15*
c^3*d^5*f^2*g^2*x + 5*c^3*d^5*f^3*g - 6*(a*c^2*d^2*g^4*x^4 + 3*a*c^2*d^2*f*g^3*x^3 + 3*a*c^2*d^2*f^2*g^2*x^2 +
a*c^2*d^2*f^3*g*x)*e^3 + (c^3*d^3*f*g^3*x^4 - 6*a*c^2*d^3*f^3*g + 3*(c^3*d^3*f^2*g^2 - 2*a*c^2*d^3*g^4)*x^3 +
3*(c^3*d^3*f^3*g - 6*a*c^2*d^3*f*g^3)*x^2 + (c^3*d^3*f^4 - 18*a*c^2*d^3*f^2*g^2)*x)*e^2 + (5*c^3*d^4*g^4*x^4
+ 16*c^3*d^4*f*g^3*x^3 + 18*c^3*d^4*f^2*g^2*x^2 + 8*c^3*d^4*f^3*g*x + c^3*d^4*f^4)*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)*sqrt(x*e + d)/(c*d^2*g*x + a*g*x*e^
2 + (c*d*g*x^2 + a*d*g)*e)) - (15*c^3*d^4*f*g^4*x^2 + 40*c^3*d^4*f^2*g^3*x + 33*c^3*d^4*f^3*g^2 - 4*(3*a^3*g^5
*x + a^3*f*g^4)*e^4 + 2*(9*a^2*c*d*g^5*x^2 + 31*a^2*c*d*f*g^4*x + 10*a^2*c*d*f^2*g^3 - 4*a^3*d*g^5)*e^3 - (21*
a*c^2*d^2*f*g^4*x^2 + 13*a*c^2*d^2*f^3*g^2 - 34*a^2*c*d^2*f*g^4 + 2*(29*a*c^2*d^2*f^2*g^3 - 5*a^2*c*d^2*g^5)*x
)*e^2 - (3*c^3*d^3*f^4*g + 59*a*c^2*d^3*f^2*g^3 - 3*(c^3*d^3*f^2*g^3 - 5*a*c^2*d^3*g^5)*x^2 - 2*(4*c^3*d^3*f^3
*g^2 - 25*a*c^2*d^3*f*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^4*d^5*f^4*g^5*x
^3 + 3*c^4*d^5*f^5*g^4*x^2 + 3*c^4*d^5*f^6*g^3*x + c^4*d^5*f^7*g^2 + (a^4*g^9*x^4 + 3*a^4*f*g^8*x^3 + 3*a^4*f^
2*g^7*x^2 + a^4*f^3*g^6*x)*e^5 - (4*a^3*c*d*f*g^8*x^4 - a^4*d*f^3*g^6 + (12*a^3*c*d*f^2*g^7 - a^4*d*g^9)*x^3 +
3*(4*a^3*c*d*f^3*g^6 - a^4*d*f*g^8)*x^2 + (4*a^3*c*d*f^4*g^5 - 3*a^4*d*f^2*g^7)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2
*g^7*x^4 - 2*a^3*c*d^2*f^4*g^5 + (9*a^2*c^2*d^2*f^3*g^6 - 2*a^3*c*d^2*f*g^8)*x^3 + 3*(3*a^2*c^2*d^2*f^4*g^5 -
2*a^3*c*d^2*f^2*g^7)*x^2 + 3*(a^2*c^2*d^2*f^5*g^4 - 2*a^3*c*d^2*f^3*g^6)*x)*e^3 - 2*(2*a*c^3*d^3*f^3*g^6*x^4 -
3*a^2*c^2*d^3*f^5*g^4 + 3*(2*a*c^3*d^3*f^4*g^5 - a^2*c^2*d^3*f^2*g^7)*x^3 + 3*(2*a*c^3*d^3*f^5*g^4 - 3*a^2*c^
2*d^3*f^3*g^6)*x^2 + (2*a*c^3*d^3*f^6*g^3 - 9*a^2*c^2*d^3*f^4*g^5)*x)*e^2 + (c^4*d^4*f^4*g^5*x^4 - 4*a*c^3*d^4
*f^6*g^3 + (3*c^4*d^4*f^5*g^4 - 4*a*c^3*d^4*f^3*g^6)*x^3 + 3*(c^4*d^4*f^6*g^3 - 4*a*c^3*d^4*f^4*g^5)*x^2 + (c^
4*d^4*f^7*g^2 - 12*a*c^3*d^4*f^5*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)**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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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)^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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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 )}^4\,\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)^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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