3.8.9 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^5} \, dx\) [709]

Optimal. Leaf size=323 \[ -\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4}+\frac {5 c^4 d^4 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{64 g^{7/2} (c d f-a e g)^{3/2}} \]

[Out]

-5/24*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g^2/(e*x+d)^(3/2)/(g*x+f)^3-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(5/2)/g/(e*x+d)^(5/2)/(g*x+f)^4+5/64*c^4*d^4*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^(7/2)/(-a*e*g+c*d*f)^(3/2)-5/32*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)/g^3/(g*x+f)^2/(e*x+d)^(1/2)+5/64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)/(g*x+f
)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {876, 886, 888, 211} \begin {gather*} \frac {5 c^4 d^4 \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 )}{64 g^{7/2} (c d f-a e g)^{3/2}}+\frac {5 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^3 \sqrt {d+e x} (f+g x) (c d f-a e g)}-\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^2}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^5),x]

[Out]

(-5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*g^3*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^3*d^3*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*g^3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)) - (5*c*d*(a*d*e + (c*
d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*g^2*(d + e*x)^(3/2)*(f + g*x)^3) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(5/2)/(4*g*(d + e*x)^(5/2)*(f + g*x)^4) + (5*c^4*d^4*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])])/(64*g^(7/2)*(c*d*f - a*e*g)^(3/2))

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 876

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*(n + 1))), x] + Dist[c*(m/(e*g*(n + 1))), Int[(d +
e*x)^(m + 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{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] &&  !IntegerQ[p] && EqQ[m + p, 0] && GtQ[p,
0] && LtQ[n, -1] &&  !(IntegerQ[n + p] && LeQ[n + p + 2, 0])

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]

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^5} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4}+\frac {(5 c d) \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)^4} \, dx}{8 g}\\ &=-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4}+\frac {\left (5 c^2 d^2\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)^3} \, dx}{16 g^2}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^2}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4}+\frac {\left (5 c^3 d^3\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}{64 g^3}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4}+\frac {\left (5 c^4 d^4\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}{128 g^3 (c d f-a e g)}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4}+\frac {\left (5 c^4 d^4 e^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 )}{64 g^3 (c d f-a e g)}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4}+\frac {5 c^4 d^4 \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 )}{64 g^{7/2} (c d f-a e g)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 1.36, size = 244, normalized size = 0.76 \begin {gather*} \frac {c^4 d^4 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {g} \left (48 a^3 e^3 g^3-8 a^2 c d e^2 g^2 (f-17 g x)+2 a c^2 d^2 e g \left (-5 f^2-18 f g x+59 g^2 x^2\right )-c^3 d^3 \left (15 f^3+55 f^2 g x+73 f g^2 x^2-15 g^3 x^3\right )\right )}{c^4 d^4 (c d f-a e g) (a e+c d x)^2 (f+g x)^4}+\frac {15 \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)^{3/2} (a e+c d x)^{5/2}}\right )}{192 g^{7/2} (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^5),x]

[Out]

(c^4*d^4*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[g]*(48*a^3*e^3*g^3 - 8*a^2*c*d*e^2*g^2*(f - 17*g*x) + 2*a*c^2*
d^2*e*g*(-5*f^2 - 18*f*g*x + 59*g^2*x^2) - c^3*d^3*(15*f^3 + 55*f^2*g*x + 73*f*g^2*x^2 - 15*g^3*x^3)))/(c^4*d^
4*(c*d*f - a*e*g)*(a*e + c*d*x)^2*(f + g*x)^4) + (15*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/
((c*d*f - a*e*g)^(3/2)*(a*e + c*d*x)^(5/2))))/(192*g^(7/2)*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(654\) vs. \(2(285)=570\).
time = 0.16, size = 655, normalized size = 2.03

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} g^{4} x^{4}+60 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f \,g^{3} x^{3}+90 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{2} g^{2} x^{2}+60 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{3} g x -15 c^{3} d^{3} g^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{4}-118 a \,c^{2} d^{2} e \,g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+73 c^{3} d^{3} f \,g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-136 a^{2} c d \,e^{2} g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+36 a \,c^{2} d^{2} e f \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+55 c^{3} d^{3} f^{2} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-48 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{3} e^{3} g^{3}+8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c d \,e^{2} f \,g^{2}+10 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e \,f^{2} g +15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f^{3}\right )}{192 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -c d f \right ) g^{3} \left (g x +f \right )^{4} \sqrt {\left (a e g -c d f \right ) g}}\)	\(655\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^5,x,method=_RETURNVERBOSE)

[Out]

1/192*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^4*d^4*g^4*x^4+60*
arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^4*d^4*f*g^3*x^3+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-
c*d*f)*g)^(1/2))*c^4*d^4*f^2*g^2*x^2+60*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^4*d^4*f^3*g*x-1
5*c^3*d^3*g^3*x^3*(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^4*d^4*f^4-118*a*c^2*d^2*e*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+73*c^3*d^3*f*g^2*x^2*(c*d*
x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-136*a^2*c*d*e^2*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+36*a*c^2*
d^2*e*f*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+55*c^3*d^3*f^2*g*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)
^(1/2)-48*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^3*e^3*g^3+8*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^
2*c*d*e^2*f*g^2+10*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a*c^2*d^2*e*f^2*g+15*(c*d*x+a*e)^(1/2)*((a*e*g-c*
d*f)*g)^(1/2)*c^3*d^3*f^3)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/(a*e*g-c*d*f)/g^3/(g*x+f)^4/((a*e*g-c*d*f)*g)^(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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^5,x, algorithm="maxima")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((g*x + f)^5*(x*e + d)^(5/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 945 vs. \(2 (298) = 596\).
time = 2.65, size = 1929, normalized size = 5.97 \begin {gather*} \left [\frac {15 \, {\left (c^{4} d^{5} g^{4} x^{4} + 4 \, c^{4} d^{5} f g^{3} x^{3} + 6 \, c^{4} d^{5} f^{2} g^{2} x^{2} + 4 \, c^{4} d^{5} f^{3} g x + c^{4} d^{5} f^{4} + {\left (c^{4} d^{4} g^{4} x^{5} + 4 \, c^{4} d^{4} f g^{3} x^{4} + 6 \, c^{4} d^{4} f^{2} g^{2} x^{3} + 4 \, c^{4} d^{4} f^{3} g x^{2} + c^{4} d^{4} f^{4} x\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 (15 \, c^{4} d^{4} f g^{4} x^{3} - 73 \, c^{4} d^{4} f^{2} g^{3} x^{2} - 55 \, c^{4} d^{4} f^{3} g^{2} x - 15 \, c^{4} d^{4} f^{4} g - 48 \, a^{4} g^{5} e^{4} - 8 \, {\left (17 \, a^{3} c d g^{5} x - 7 \, a^{3} c d f g^{4}\right )} e^{3} - 2 \, {\left (59 \, a^{2} c^{2} d^{2} g^{5} x^{2} - 86 \, a^{2} c^{2} d^{2} f g^{4} x - a^{2} c^{2} d^{2} f^{2} g^{3}\right )} e^{2} - {\left (15 \, a c^{3} d^{3} g^{5} x^{3} - 191 \, a c^{3} d^{3} f g^{4} x^{2} - 19 \, a c^{3} d^{3} f^{2} g^{3} x - 5 \, a c^{3} d^{3} f^{3} g^{2}\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}}{384 \, {\left (c^{2} d^{3} f^{2} g^{8} x^{4} + 4 \, c^{2} d^{3} f^{3} g^{7} x^{3} + 6 \, c^{2} d^{3} f^{4} g^{6} x^{2} + 4 \, c^{2} d^{3} f^{5} g^{5} x + c^{2} d^{3} f^{6} g^{4} + {\left (a^{2} g^{10} x^{5} + 4 \, a^{2} f g^{9} x^{4} + 6 \, a^{2} f^{2} g^{8} x^{3} + 4 \, a^{2} f^{3} g^{7} x^{2} + a^{2} f^{4} g^{6} x\right )} e^{3} - {\left (2 \, a c d f g^{9} x^{5} - a^{2} d f^{4} g^{6} + {\left (8 \, a c d f^{2} g^{8} - a^{2} d g^{10}\right )} x^{4} + 4 \, {\left (3 \, a c d f^{3} g^{7} - a^{2} d f g^{9}\right )} x^{3} + 2 \, {\left (4 \, a c d f^{4} g^{6} - 3 \, a^{2} d f^{2} g^{8}\right )} x^{2} + 2 \, {\left (a c d f^{5} g^{5} - 2 \, a^{2} d f^{3} g^{7}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{8} x^{5} - 2 \, a c d^{2} f^{5} g^{5} + 2 \, {\left (2 \, c^{2} d^{2} f^{3} g^{7} - a c d^{2} f g^{9}\right )} x^{4} + 2 \, {\left (3 \, c^{2} d^{2} f^{4} g^{6} - 4 \, a c d^{2} f^{2} g^{8}\right )} x^{3} + 4 \, {\left (c^{2} d^{2} f^{5} g^{5} - 3 \, a c d^{2} f^{3} g^{7}\right )} x^{2} + {\left (c^{2} d^{2} f^{6} g^{4} - 8 \, a c d^{2} f^{4} g^{6}\right )} x\right )} e\right )}}, -\frac {15 \, {\left (c^{4} d^{5} g^{4} x^{4} + 4 \, c^{4} d^{5} f g^{3} x^{3} + 6 \, c^{4} d^{5} f^{2} g^{2} x^{2} + 4 \, c^{4} d^{5} f^{3} g x + c^{4} d^{5} f^{4} + {\left (c^{4} d^{4} g^{4} x^{5} + 4 \, c^{4} d^{4} f g^{3} x^{4} + 6 \, c^{4} d^{4} f^{2} g^{2} x^{3} + 4 \, c^{4} d^{4} f^{3} g x^{2} + c^{4} d^{4} f^{4} x\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 (15 \, c^{4} d^{4} f g^{4} x^{3} - 73 \, c^{4} d^{4} f^{2} g^{3} x^{2} - 55 \, c^{4} d^{4} f^{3} g^{2} x - 15 \, c^{4} d^{4} f^{4} g - 48 \, a^{4} g^{5} e^{4} - 8 \, {\left (17 \, a^{3} c d g^{5} x - 7 \, a^{3} c d f g^{4}\right )} e^{3} - 2 \, {\left (59 \, a^{2} c^{2} d^{2} g^{5} x^{2} - 86 \, a^{2} c^{2} d^{2} f g^{4} x - a^{2} c^{2} d^{2} f^{2} g^{3}\right )} e^{2} - {\left (15 \, a c^{3} d^{3} g^{5} x^{3} - 191 \, a c^{3} d^{3} f g^{4} x^{2} - 19 \, a c^{3} d^{3} f^{2} g^{3} x - 5 \, a c^{3} d^{3} f^{3} g^{2}\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}}{192 \, {\left (c^{2} d^{3} f^{2} g^{8} x^{4} + 4 \, c^{2} d^{3} f^{3} g^{7} x^{3} + 6 \, c^{2} d^{3} f^{4} g^{6} x^{2} + 4 \, c^{2} d^{3} f^{5} g^{5} x + c^{2} d^{3} f^{6} g^{4} + {\left (a^{2} g^{10} x^{5} + 4 \, a^{2} f g^{9} x^{4} + 6 \, a^{2} f^{2} g^{8} x^{3} + 4 \, a^{2} f^{3} g^{7} x^{2} + a^{2} f^{4} g^{6} x\right )} e^{3} - {\left (2 \, a c d f g^{9} x^{5} - a^{2} d f^{4} g^{6} + {\left (8 \, a c d f^{2} g^{8} - a^{2} d g^{10}\right )} x^{4} + 4 \, {\left (3 \, a c d f^{3} g^{7} - a^{2} d f g^{9}\right )} x^{3} + 2 \, {\left (4 \, a c d f^{4} g^{6} - 3 \, a^{2} d f^{2} g^{8}\right )} x^{2} + 2 \, {\left (a c d f^{5} g^{5} - 2 \, a^{2} d f^{3} g^{7}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{8} x^{5} - 2 \, a c d^{2} f^{5} g^{5} + 2 \, {\left (2 \, c^{2} d^{2} f^{3} g^{7} - a c d^{2} f g^{9}\right )} x^{4} + 2 \, {\left (3 \, c^{2} d^{2} f^{4} g^{6} - 4 \, a c d^{2} f^{2} g^{8}\right )} x^{3} + 4 \, {\left (c^{2} d^{2} f^{5} g^{5} - 3 \, a c d^{2} f^{3} g^{7}\right )} x^{2} + {\left (c^{2} d^{2} f^{6} g^{4} - 8 \, a c d^{2} f^{4} g^{6}\right )} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^5,x, algorithm="fricas")

[Out]

[1/384*(15*(c^4*d^5*g^4*x^4 + 4*c^4*d^5*f*g^3*x^3 + 6*c^4*d^5*f^2*g^2*x^2 + 4*c^4*d^5*f^3*g*x + c^4*d^5*f^4 +
(c^4*d^4*g^4*x^5 + 4*c^4*d^4*f*g^3*x^4 + 6*c^4*d^4*f^2*g^2*x^3 + 4*c^4*d^4*f^3*g*x^2 + c^4*d^4*f^4*x)*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^4*d^4*f*g^4*x^3 - 73*c^4*d^4*f^2*g^3*x^2 - 55*c^4*d^4*f^3*g^2*x - 15*c^4*d^4*f^4*g - 48*a^4*g^5*e^4 - 8
*(17*a^3*c*d*g^5*x - 7*a^3*c*d*f*g^4)*e^3 - 2*(59*a^2*c^2*d^2*g^5*x^2 - 86*a^2*c^2*d^2*f*g^4*x - a^2*c^2*d^2*f
^2*g^3)*e^2 - (15*a*c^3*d^3*g^5*x^3 - 191*a*c^3*d^3*f*g^4*x^2 - 19*a*c^3*d^3*f^2*g^3*x - 5*a*c^3*d^3*f^3*g^2)*
e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^2*d^3*f^2*g^8*x^4 + 4*c^2*d^3*f^3*g^7*x^3 + 6
*c^2*d^3*f^4*g^6*x^2 + 4*c^2*d^3*f^5*g^5*x + c^2*d^3*f^6*g^4 + (a^2*g^10*x^5 + 4*a^2*f*g^9*x^4 + 6*a^2*f^2*g^8
*x^3 + 4*a^2*f^3*g^7*x^2 + a^2*f^4*g^6*x)*e^3 - (2*a*c*d*f*g^9*x^5 - a^2*d*f^4*g^6 + (8*a*c*d*f^2*g^8 - a^2*d*
g^10)*x^4 + 4*(3*a*c*d*f^3*g^7 - a^2*d*f*g^9)*x^3 + 2*(4*a*c*d*f^4*g^6 - 3*a^2*d*f^2*g^8)*x^2 + 2*(a*c*d*f^5*g
^5 - 2*a^2*d*f^3*g^7)*x)*e^2 + (c^2*d^2*f^2*g^8*x^5 - 2*a*c*d^2*f^5*g^5 + 2*(2*c^2*d^2*f^3*g^7 - a*c*d^2*f*g^9
)*x^4 + 2*(3*c^2*d^2*f^4*g^6 - 4*a*c*d^2*f^2*g^8)*x^3 + 4*(c^2*d^2*f^5*g^5 - 3*a*c*d^2*f^3*g^7)*x^2 + (c^2*d^2
*f^6*g^4 - 8*a*c*d^2*f^4*g^6)*x)*e), -1/192*(15*(c^4*d^5*g^4*x^4 + 4*c^4*d^5*f*g^3*x^3 + 6*c^4*d^5*f^2*g^2*x^2
+ 4*c^4*d^5*f^3*g*x + c^4*d^5*f^4 + (c^4*d^4*g^4*x^5 + 4*c^4*d^4*f*g^3*x^4 + 6*c^4*d^4*f^2*g^2*x^3 + 4*c^4*d^
4*f^3*g*x^2 + c^4*d^4*f^4*x)*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^4*d^4*f*g^4*x^3 -
73*c^4*d^4*f^2*g^3*x^2 - 55*c^4*d^4*f^3*g^2*x - 15*c^4*d^4*f^4*g - 48*a^4*g^5*e^4 - 8*(17*a^3*c*d*g^5*x - 7*a^
3*c*d*f*g^4)*e^3 - 2*(59*a^2*c^2*d^2*g^5*x^2 - 86*a^2*c^2*d^2*f*g^4*x - a^2*c^2*d^2*f^2*g^3)*e^2 - (15*a*c^3*d
^3*g^5*x^3 - 191*a*c^3*d^3*f*g^4*x^2 - 19*a*c^3*d^3*f^2*g^3*x - 5*a*c^3*d^3*f^3*g^2)*e)*sqrt(c*d^2*x + a*x*e^2
+ (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^2*d^3*f^2*g^8*x^4 + 4*c^2*d^3*f^3*g^7*x^3 + 6*c^2*d^3*f^4*g^6*x^2 + 4*
c^2*d^3*f^5*g^5*x + c^2*d^3*f^6*g^4 + (a^2*g^10*x^5 + 4*a^2*f*g^9*x^4 + 6*a^2*f^2*g^8*x^3 + 4*a^2*f^3*g^7*x^2
+ a^2*f^4*g^6*x)*e^3 - (2*a*c*d*f*g^9*x^5 - a^2*d*f^4*g^6 + (8*a*c*d*f^2*g^8 - a^2*d*g^10)*x^4 + 4*(3*a*c*d*f^
3*g^7 - a^2*d*f*g^9)*x^3 + 2*(4*a*c*d*f^4*g^6 - 3*a^2*d*f^2*g^8)*x^2 + 2*(a*c*d*f^5*g^5 - 2*a^2*d*f^3*g^7)*x)*
e^2 + (c^2*d^2*f^2*g^8*x^5 - 2*a*c*d^2*f^5*g^5 + 2*(2*c^2*d^2*f^3*g^7 - a*c*d^2*f*g^9)*x^4 + 2*(3*c^2*d^2*f^4*
g^6 - 4*a*c*d^2*f^2*g^8)*x^3 + 4*(c^2*d^2*f^5*g^5 - 3*a*c*d^2*f^3*g^7)*x^2 + (c^2*d^2*f^6*g^4 - 8*a*c*d^2*f^4*
g^6)*x)*e)]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**5,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 deep

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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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^5,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 (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (f+g\,x\right )}^5\,{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^5*(d + e*x)^(5/2)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^5*(d + e*x)^(5/2)), x)

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