3.20.37 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^2} \, dx\) [1937]

Optimal. Leaf size=261 \[ \frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d e^3}+\frac {5}{24} \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{7/2}} \]

[Out]

5/24*(a-c*d^2/e^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/4*(c*d*x+a*e)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)/e-5/128*(-a*e^2+c*d^2)^4*arctanh(1/2*(2*c*d*e*x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2))/c^(3/2)/d^(3/2)/e^(7/2)+5/64*(-a*e^2+c*d^2)^2*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)/c/d/e^3

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Rubi [A]
time = 0.14, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {668, 684, 654, 626, 635, 212} \begin {gather*} -\frac {5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{7/2}}+\frac {5}{24} \left (a-\frac {c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}+\frac {5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c d e^3} \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)^2,x]

[Out]

(5*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*c*d*e^3) + (
5*(a - (c*d^2)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/24 + ((a*e + c*d*x)*(a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2)^(3/2))/(4*e) - (5*(c*d^2 - a*e^2)^4*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sq
rt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(128*c^(3/2)*d^(3/2)*e^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 668

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(a + b*x + c*x^2)^(m +
p)/(a/d + c*(x/e))^m, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IntegerQ[p] && IntegerQ[m] && RationalQ[p] && (LtQ[0, -m, p] || LtQ[p, -m, 0]) && NeQ[m, 2] && NeQ[m, -1
]

Rule 684

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

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)^2} \, dx &=\int (a e+c d x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\\ &=\frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}+\frac {\left (5 \left (2 a c d e^2-c d \left (c d^2+a e^2\right )\right )\right ) \int (a e+c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{8 c d e}\\ &=\frac {5}{24} \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}+\frac {\left (5 \left (c d^2-a e^2\right )^2\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{16 e^2}\\ &=\frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d e^3}+\frac {5}{24} \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}-\frac {\left (5 \left (c d^2-a e^2\right )^4\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c d e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d e^3}+\frac {5}{24} \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}-\frac {\left (5 \left (c d^2-a e^2\right )^4\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 c d e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d e^3}+\frac {5}{24} \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 236, normalized size = 0.90 \begin {gather*} \frac {\left (c d^2-a e^2\right )^4 \sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (15 a^3 e^6+a^2 c d e^4 (73 d+118 e x)+a c^2 d^2 e^2 \left (-55 d^2+36 d e x+136 e^2 x^2\right )+c^3 d^3 \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^4}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 c^{3/2} d^{3/2} e^{7/2}} \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)^2,x]

[Out]

((c*d^2 - a*e^2)^4*Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(15*a^3*e^6 + a^2*c*d*e^4*(73*d + 1
18*e*x) + a*c^2*d^2*e^2*(-55*d^2 + 36*d*e*x + 136*e^2*x^2) + c^3*d^3*(15*d^3 - 10*d^2*e*x + 8*d*e^2*x^2 + 48*e
^3*x^3)))/(c*d^2 - a*e^2)^4 - (15*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[
a*e + c*d*x]*Sqrt[d + e*x])))/(192*c^(3/2)*d^(3/2)*e^(7/2))

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Maple [A]
time = 0.70, size = 408, normalized size = 1.56

method	result	size
default	\(\frac {\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {10 c d e \left (\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c d e}-\frac {3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )}}{e^{2}}\)	\(408\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

1/e^2*(2/3/(a*e^2-c*d^2)/(x+d/e)^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(7/2)-10/3*c*d*e/(a*e^2-c*d^2)*(1/5
*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)+1/2*(a*e^2-c*d^2)*(1/8*(2*c*d*e*(x+d/e)+e^2*a-c*d^2)/c/d/e*(c*d
*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/16*(a*e^2-c*d^2)^2/c/d/e*(1/4*(2*c*d*e*(x+d/e)+e^2*a-c*d^2)/c/d/e*
(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/c/d/e*ln((1/2*e^2*a-1/2*c*d^2+c*d*e*(x+d/e))
/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (230) = 460\).
time = 0.30, size = 709, normalized size = 2.72 \begin {gather*} -\frac {5 \, c^{4} d^{8} e^{\left (-\frac {7}{2}\right )} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{128 \, \left (c d\right )^{\frac {3}{2}}} + \frac {5 \, a c^{3} d^{6} e^{\left (-\frac {3}{2}\right )} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{32 \, \left (c d\right )^{\frac {3}{2}}} + \frac {5}{32} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} c^{2} d^{4} x e^{\left (-2\right )} + \frac {5}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} c^{2} d^{5} e^{\left (-3\right )} - \frac {15 \, a^{2} c^{2} d^{4} e^{\frac {1}{2}} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{64 \, \left (c d\right )^{\frac {3}{2}}} - \frac {5}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a c d^{3} e^{\left (-1\right )} + \frac {5 \, a^{3} c d^{2} e^{\frac {5}{2}} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{32 \, \left (c d\right )^{\frac {3}{2}}} - \frac {5}{16} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a c d^{2} x - \frac {5}{24} \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}} c d^{2} e^{\left (-2\right )} + \frac {5}{32} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{2} x e^{2} - \frac {5}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{2} d e - \frac {5 \, a^{4} e^{\frac {9}{2}} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{128 \, \left (c d\right )^{\frac {3}{2}}} + \frac {5}{24} \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}} a + \frac {5 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{3} e^{3}}{64 \, c d} + \frac {{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {5}{2}}}{4 \, {\left (x e^{2} + d e\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)^2,x, algorithm="maxima")

[Out]

-5/128*c^4*d^8*e^(-7/2)*log(c*d^2*e^(-1) + 2*c*d*x + a*e + 2*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*sqrt(
c*d)*e^(-1/2))/(c*d)^(3/2) + 5/32*a*c^3*d^6*e^(-3/2)*log(c*d^2*e^(-1) + 2*c*d*x + a*e + 2*sqrt(c*d*x^2*e + c*d
^2*x + a*x*e^2 + a*d*e)*sqrt(c*d)*e^(-1/2))/(c*d)^(3/2) + 5/32*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*c^2
*d^4*x*e^(-2) + 5/64*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*c^2*d^5*e^(-3) - 15/64*a^2*c^2*d^4*e^(1/2)*lo
g(c*d^2*e^(-1) + 2*c*d*x + a*e + 2*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*sqrt(c*d)*e^(-1/2))/(c*d)^(3/2)
- 5/64*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*a*c*d^3*e^(-1) + 5/32*a^3*c*d^2*e^(5/2)*log(c*d^2*e^(-1) +
2*c*d*x + a*e + 2*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*sqrt(c*d)*e^(-1/2))/(c*d)^(3/2) - 5/16*sqrt(c*d
*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*a*c*d^2*x - 5/24*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^(3/2)*c*d^2*e^(-2
) + 5/32*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*a^2*x*e^2 - 5/64*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d
*e)*a^2*d*e - 5/128*a^4*e^(9/2)*log(c*d^2*e^(-1) + 2*c*d*x + a*e + 2*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*
e)*sqrt(c*d)*e^(-1/2))/(c*d)^(3/2) + 5/24*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^(3/2)*a + 5/64*sqrt(c*d*x^2*
e + c*d^2*x + a*x*e^2 + a*d*e)*a^3*e^3/(c*d) + 1/4*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^(5/2)/(x*e^2 + d*e)

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Fricas [A]
time = 3.03, size = 651, normalized size = 2.49 \begin {gather*} \left [\frac {{\left (15 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {c d} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} - 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) - 4 \, {\left (10 \, c^{4} d^{6} x e^{2} - 15 \, c^{4} d^{7} e - 118 \, a^{2} c^{2} d^{2} x e^{6} - 15 \, a^{3} c d e^{7} - {\left (136 \, a c^{3} d^{3} x^{2} + 73 \, a^{2} c^{2} d^{3}\right )} e^{5} - 12 \, {\left (4 \, c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{4} x\right )} e^{4} - {\left (8 \, c^{4} d^{5} x^{2} - 55 \, a c^{3} d^{5}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{768 \, c^{2} d^{2}}, \frac {{\left (15 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (10 \, c^{4} d^{6} x e^{2} - 15 \, c^{4} d^{7} e - 118 \, a^{2} c^{2} d^{2} x e^{6} - 15 \, a^{3} c d e^{7} - {\left (136 \, a c^{3} d^{3} x^{2} + 73 \, a^{2} c^{2} d^{3}\right )} e^{5} - 12 \, {\left (4 \, c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{4} x\right )} e^{4} - {\left (8 \, c^{4} d^{5} x^{2} - 55 \, a c^{3} d^{5}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{384 \, c^{2} d^{2}}\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)^2,x, algorithm="fricas")

[Out]

[1/768*(15*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(c*d)*e^(1/2)*log(8
*c^2*d^3*x*e + c^2*d^4 + 8*a*c*d*x*e^3 + a^2*e^4 - 4*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e +
c*d^2 + a*e^2)*sqrt(c*d)*e^(1/2) + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2) - 4*(10*c^4*d^6*x*e^2 - 15*c^4*d^7*e - 1
18*a^2*c^2*d^2*x*e^6 - 15*a^3*c*d*e^7 - (136*a*c^3*d^3*x^2 + 73*a^2*c^2*d^3)*e^5 - 12*(4*c^4*d^4*x^3 + 3*a*c^3
*d^4*x)*e^4 - (8*c^4*d^5*x^2 - 55*a*c^3*d^5)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-4)/(c^2*d^2
), 1/384*(15*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(-c*d*e)*arctan(1
/2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^3*x*e + a*c*d*x
*e^3 + (c^2*d^2*x^2 + a*c*d^2)*e^2)) - 2*(10*c^4*d^6*x*e^2 - 15*c^4*d^7*e - 118*a^2*c^2*d^2*x*e^6 - 15*a^3*c*d
*e^7 - (136*a*c^3*d^3*x^2 + 73*a^2*c^2*d^3)*e^5 - 12*(4*c^4*d^4*x^3 + 3*a*c^3*d^4*x)*e^4 - (8*c^4*d^5*x^2 - 55
*a*c^3*d^5)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-4)/(c^2*d^2)]

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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)**(5/2)/(e*x+d)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1330 vs. \(2 (230) = 460\).
time = 1.60, size = 1330, normalized size = 5.10 \begin {gather*} \frac {1}{192} \, {\left (\frac {15 \, {\left (c^{4} d^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 4 \, a c^{3} d^{6} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 6 \, a^{2} c^{2} d^{4} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 4 \, a^{3} c d^{2} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + a^{4} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} \arctan \left (\frac {\sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}}}{\sqrt {-c d e}}\right ) e^{\left (-4\right )}}{\sqrt {-c d e} c d} + \frac {{\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{7} d^{11} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 55 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c^{6} d^{10} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 60 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a c^{6} d^{9} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 73 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} c^{5} d^{9} e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 220 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a c^{5} d^{8} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 15 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} c^{4} d^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 90 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a^{2} c^{5} d^{7} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 292 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} a c^{4} d^{7} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 330 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a^{2} c^{4} d^{6} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 60 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} a c^{3} d^{6} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 60 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a^{3} c^{4} d^{5} e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 438 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} a^{2} c^{3} d^{5} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 220 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a^{3} c^{3} d^{4} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 90 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} a^{2} c^{2} d^{4} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 15 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a^{4} c^{3} d^{3} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 292 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} a^{3} c^{2} d^{3} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 55 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a^{4} c^{2} d^{2} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 60 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} a^{3} c d^{2} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 73 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {5}{2}} a^{4} c d e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 15 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {7}{2}} a^{4} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} e^{\left (-4\right )}}{{\left (\frac {c d^{2} e}{x e + d} - \frac {a e^{3}}{x e + d}\right )}^{4} c d}\right )} e \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)^2,x, algorithm="giac")

[Out]

1/192*(15*(c^4*d^8*sgn(1/(x*e + d)) - 4*a*c^3*d^6*e^2*sgn(1/(x*e + d)) + 6*a^2*c^2*d^4*e^4*sgn(1/(x*e + d)) -
4*a^3*c*d^2*e^6*sgn(1/(x*e + d)) + a^4*e^8*sgn(1/(x*e + d)))*arctan(sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*
e + d))/sqrt(-c*d*e))*e^(-4)/(sqrt(-c*d*e)*c*d) + (15*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*c^7*d^
11*e^3*sgn(1/(x*e + d)) - 55*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*c^6*d^10*e^2*sgn(1/(x*e + d))
- 60*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a*c^6*d^9*e^5*sgn(1/(x*e + d)) + 73*(c*d*e - c*d^2*e/(
x*e + d) + a*e^3/(x*e + d))^(5/2)*c^5*d^9*e*sgn(1/(x*e + d)) + 220*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d
))^(3/2)*a*c^5*d^8*e^4*sgn(1/(x*e + d)) + 15*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/2)*c^4*d^8*sgn(1
/(x*e + d)) + 90*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^2*c^5*d^7*e^7*sgn(1/(x*e + d)) - 292*(c*d
*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a*c^4*d^7*e^3*sgn(1/(x*e + d)) - 330*(c*d*e - c*d^2*e/(x*e + d
) + a*e^3/(x*e + d))^(3/2)*a^2*c^4*d^6*e^6*sgn(1/(x*e + d)) - 60*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))
^(7/2)*a*c^3*d^6*e^2*sgn(1/(x*e + d)) - 60*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^3*c^4*d^5*e^9*s
gn(1/(x*e + d)) + 438*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^2*c^3*d^5*e^5*sgn(1/(x*e + d)) + 2
20*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^3*c^3*d^4*e^8*sgn(1/(x*e + d)) + 90*(c*d*e - c*d^2*e/
(x*e + d) + a*e^3/(x*e + d))^(7/2)*a^2*c^2*d^4*e^4*sgn(1/(x*e + d)) + 15*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^
3/(x*e + d))*a^4*c^3*d^3*e^11*sgn(1/(x*e + d)) - 292*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^3*c
^2*d^3*e^7*sgn(1/(x*e + d)) - 55*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^4*c^2*d^2*e^10*sgn(1/(x
*e + d)) - 60*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/2)*a^3*c*d^2*e^6*sgn(1/(x*e + d)) + 73*(c*d*e -
c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^4*c*d*e^9*sgn(1/(x*e + d)) + 15*(c*d*e - c*d^2*e/(x*e + d) + a*e
^3/(x*e + d))^(7/2)*a^4*e^8*sgn(1/(x*e + d)))*e^(-4)/((c*d^2*e/(x*e + d) - a*e^3/(x*e + d))^4*c*d))*e

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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 (d+e\,x\right )}^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)/(d + e*x)^2,x)

[Out]

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

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