3.7.88 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^5} \, dx\) [688]

Optimal. Leaf size=347 \[ -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {5 c^4 d^4 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{64 g^{3/2} (c d f-a e g)^{7/2}} \]

[Out]

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^(3/2
)/(-a*e*g+c*d*f)^(7/2)-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(g*x+f)^4/(e*x+d)^(1/2)+1/24*c*d*(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)+5/96*c^2*d^2*(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)+5/64*c^3*d^3*(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 = 0.32, antiderivative size = 347, 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^{3/2} (c d f-a e g)^{7/2}}+\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 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 g \sqrt {d+e x} (f+g x)^3 (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}}{4 g \sqrt {d+e x} (f+g x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(g*Sqrt[d + e*x]*(f + g*x)^4) + (c*d*Sqrt[a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2])/(24*g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) + (5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2])/(96*g*(c*d*f - a*e*g)^2*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*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (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^(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 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 {\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)^5} \, dx &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {(c d) \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {\left (5 c^2 d^2\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}{48 g (c d f-a e g)}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^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}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\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 (c d f-a e g)^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^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}}{96 g (c d f-a e g)^2 \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 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\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 (c d f-a e g)^3}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^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}}{96 g (c d f-a e g)^2 \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 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\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 (c d f-a e g)^3}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^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}}{96 g (c d f-a e g)^2 \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 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\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^{3/2} (c d f-a e g)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.20, size = 234, normalized size = 0.67 \begin {gather*} \frac {c^4 d^4 \sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {g} \left (48 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (-17 f+g x)-2 a c^2 d^2 e g \left (-59 f^2+18 f g x+5 g^2 x^2\right )+c^3 d^3 \left (-15 f^3+73 f^2 g x+55 f g^2 x^2+15 g^3 x^3\right )\right )}{c^4 d^4 (c d f-a e g)^3 (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)^{7/2} \sqrt {a e+c d x}}\right )}{192 g^{3/2} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(685\) vs. \(2(309)=618\).
time = 0.14, size = 686, normalized size = 1.98

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} g^{4} x^{4}+60 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f \,g^{3} x^{3}+90 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{2} g^{2} x^{2}+60 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{3} g x -15 c^{3} d^{3} g^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{4}+10 a \,c^{2} d^{2} e \,g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-55 c^{3} d^{3} f \,g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-8 a^{2} c d \,e^{2} g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+36 a \,c^{2} d^{2} e f \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-73 c^{3} d^{3} f^{2} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-48 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{3} e^{3} g^{3}+136 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c d \,e^{2} f \,g^{2}-118 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e \,f^{2} g +15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f^{3}\right )}{192 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{4} g \left (a e g -c d f \right ) \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {c d x +a e}}\)	\(686\)

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)^(1/2)/(g*x+f)^5/(e*x+d)^(1/2),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+10*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)-55*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)-8*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)-73*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+136*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^2
*c*d*e^2*f*g^2-118*(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)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^4/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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^5/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1338 vs. \(2 (324) = 648\).
time = 3.52, size = 2715, normalized size = 7.82 \begin {gather*} \text {Too large to display} \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)^(1/2)/(g*x+f)^5/(e*x+d)^(1/2),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 + 55*c^4*d^4*f^2*g^3*x^2 + 73*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
*(a^3*c*d*g^5*x - 23*a^3*c*d*f*g^4)*e^3 + 2*(5*a^2*c^2*d^2*g^5*x^2 + 22*a^2*c^2*d^2*f*g^4*x - 127*a^2*c^2*d^2*
f^2*g^3)*e^2 - (15*a*c^3*d^3*g^5*x^3 + 65*a*c^3*d^3*f*g^4*x^2 + 109*a*c^3*d^3*f^2*g^3*x - 133*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^4*d^5*f^4*g^6*x^4 + 4*c^4*d^5*f^5*g^5*x^3
+ 6*c^4*d^5*f^6*g^4*x^2 + 4*c^4*d^5*f^7*g^3*x + c^4*d^5*f^8*g^2 + (a^4*g^10*x^5 + 4*a^4*f*g^9*x^4 + 6*a^4*f^2*
g^8*x^3 + 4*a^4*f^3*g^7*x^2 + a^4*f^4*g^6*x)*e^5 - (4*a^3*c*d*f*g^9*x^5 - a^4*d*f^4*g^6 + (16*a^3*c*d*f^2*g^8
- a^4*d*g^10)*x^4 + 4*(6*a^3*c*d*f^3*g^7 - a^4*d*f*g^9)*x^3 + 2*(8*a^3*c*d*f^4*g^6 - 3*a^4*d*f^2*g^8)*x^2 + 4*
(a^3*c*d*f^5*g^5 - a^4*d*f^3*g^7)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^8*x^5 - 2*a^3*c*d^2*f^5*g^5 + 2*(6*a^2*c^2*d
^2*f^3*g^7 - a^3*c*d^2*f*g^9)*x^4 + 2*(9*a^2*c^2*d^2*f^4*g^6 - 4*a^3*c*d^2*f^2*g^8)*x^3 + 12*(a^2*c^2*d^2*f^5*
g^5 - a^3*c*d^2*f^3*g^7)*x^2 + (3*a^2*c^2*d^2*f^6*g^4 - 8*a^3*c*d^2*f^4*g^6)*x)*e^3 - 2*(2*a*c^3*d^3*f^3*g^7*x
^5 - 3*a^2*c^2*d^3*f^6*g^4 + (8*a*c^3*d^3*f^4*g^6 - 3*a^2*c^2*d^3*f^2*g^8)*x^4 + 12*(a*c^3*d^3*f^5*g^5 - a^2*c
^2*d^3*f^3*g^7)*x^3 + 2*(4*a*c^3*d^3*f^6*g^4 - 9*a^2*c^2*d^3*f^4*g^6)*x^2 + 2*(a*c^3*d^3*f^7*g^3 - 6*a^2*c^2*d
^3*f^5*g^5)*x)*e^2 + (c^4*d^4*f^4*g^6*x^5 - 4*a*c^3*d^4*f^7*g^3 + 4*(c^4*d^4*f^5*g^5 - a*c^3*d^4*f^3*g^7)*x^4
+ 2*(3*c^4*d^4*f^6*g^4 - 8*a*c^3*d^4*f^4*g^6)*x^3 + 4*(c^4*d^4*f^7*g^3 - 6*a*c^3*d^4*f^5*g^5)*x^2 + (c^4*d^4*f
^8*g^2 - 16*a*c^3*d^4*f^6*g^4)*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 +
55*c^4*d^4*f^2*g^3*x^2 + 73*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*(a^3*c*d*g^5*x - 23*a^3
*c*d*f*g^4)*e^3 + 2*(5*a^2*c^2*d^2*g^5*x^2 + 22*a^2*c^2*d^2*f*g^4*x - 127*a^2*c^2*d^2*f^2*g^3)*e^2 - (15*a*c^3
*d^3*g^5*x^3 + 65*a*c^3*d^3*f*g^4*x^2 + 109*a*c^3*d^3*f^2*g^3*x - 133*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^4*d^5*f^4*g^6*x^4 + 4*c^4*d^5*f^5*g^5*x^3 + 6*c^4*d^5*f^6*g^4*x^2
+ 4*c^4*d^5*f^7*g^3*x + c^4*d^5*f^8*g^2 + (a^4*g^10*x^5 + 4*a^4*f*g^9*x^4 + 6*a^4*f^2*g^8*x^3 + 4*a^4*f^3*g^7*
x^2 + a^4*f^4*g^6*x)*e^5 - (4*a^3*c*d*f*g^9*x^5 - a^4*d*f^4*g^6 + (16*a^3*c*d*f^2*g^8 - a^4*d*g^10)*x^4 + 4*(6
*a^3*c*d*f^3*g^7 - a^4*d*f*g^9)*x^3 + 2*(8*a^3*c*d*f^4*g^6 - 3*a^4*d*f^2*g^8)*x^2 + 4*(a^3*c*d*f^5*g^5 - a^4*d
*f^3*g^7)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^8*x^5 - 2*a^3*c*d^2*f^5*g^5 + 2*(6*a^2*c^2*d^2*f^3*g^7 - a^3*c*d^2*f
*g^9)*x^4 + 2*(9*a^2*c^2*d^2*f^4*g^6 - 4*a^3*c*d^2*f^2*g^8)*x^3 + 12*(a^2*c^2*d^2*f^5*g^5 - a^3*c*d^2*f^3*g^7)
*x^2 + (3*a^2*c^2*d^2*f^6*g^4 - 8*a^3*c*d^2*f^4*g^6)*x)*e^3 - 2*(2*a*c^3*d^3*f^3*g^7*x^5 - 3*a^2*c^2*d^3*f^6*g
^4 + (8*a*c^3*d^3*f^4*g^6 - 3*a^2*c^2*d^3*f^2*g^8)*x^4 + 12*(a*c^3*d^3*f^5*g^5 - a^2*c^2*d^3*f^3*g^7)*x^3 + 2*
(4*a*c^3*d^3*f^6*g^4 - 9*a^2*c^2*d^3*f^4*g^6)*x^2 + 2*(a*c^3*d^3*f^7*g^3 - 6*a^2*c^2*d^3*f^5*g^5)*x)*e^2 + (c^
4*d^4*f^4*g^6*x^5 - 4*a*c^3*d^4*f^7*g^3 + 4*(c^4*d^4*f^5*g^5 - a*c^3*d^4*f^3*g^7)*x^4 + 2*(3*c^4*d^4*f^6*g^4 -
8*a*c^3*d^4*f^4*g^6)*x^3 + 4*(c^4*d^4*f^7*g^3 - 6*a*c^3*d^4*f^5*g^5)*x^2 + (c^4*d^4*f^8*g^2 - 16*a*c^3*d^4*f^
6*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**5/(e*x+d)**(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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^5/(e*x+d)^(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 {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (f+g\,x\right )}^5\,\sqrt {d+e\,x}} \,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)^(1/2)/((f + g*x)^5*(d + e*x)^(1/2)),x)

[Out]

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

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