3.5.60 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{d+e x} \, dx\) [460]

Optimal. Leaf size=274 \[ \frac {3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}} \]

[Out]

1/16*(a/c/d-d/e^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)^(3/2)+1/5*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(5/2)/e-3/256*(-a*e^2+c*d^2)^5*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^(5/2)/d^(5/2)/e^(7/2)+3/128*(-a*e^2+c*d^2)^3*(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^2/d^2/e^3

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Rubi [A]
time = 0.11, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {678, 626, 635, 212} \begin {gather*} -\frac {3 \left (c d^2-a e^2\right )^5 \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 )}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac {3 \left (c d^2-a e^2\right )^3 \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}}{128 c^2 d^2 e^3}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \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),x]

[Out]

(3*(c*d^2 - a*e^2)^3*(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])/(128*c^2*d^2*e^3
) + ((a/(c*d) - d/e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/16 + (a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*e) - (3*(c*d^2 - a*e^2)^5*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqr
t[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(256*c^(5/2)*d^(5/2)*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 678

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], 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] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && 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} \, dx &=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (2 c d^2 e-e \left (c d^2+a e^2\right )\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{2 e^2}\\ &=\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}+\frac {\left (3 \left (c d^2-a e^2\right )^3\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 c d e^2}\\ &=\frac {3 \left (c d^2-a e^2\right )^3 \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}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (3 \left (c d^2-a e^2\right )^5\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^2 d^2 e^3}\\ &=\frac {3 \left (c d^2-a e^2\right )^3 \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}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (3 \left (c d^2-a e^2\right )^5\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 )}{128 c^2 d^2 e^3}\\ &=\frac {3 \left (c d^2-a e^2\right )^3 \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}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {3 \left (c d^2-a e^2\right )^5 \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 )}{256 c^{5/2} d^{5/2} e^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 267, normalized size = 0.97 \begin {gather*} \frac {\left (c d^2-a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} (d+e x)^3 \left (15 c^4 d^4-\frac {15 e^4 (a e+c d x)^4}{(d+e x)^4}+\frac {70 c d e^3 (a e+c d x)^3}{(d+e x)^3}+\frac {128 c^2 d^2 e^2 (a e+c d x)^2}{(d+e x)^2}-\frac {70 c^3 d^3 e (a e+c d x)}{d+e x}\right )}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{640 c^{5/2} d^{5/2} 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),x]

[Out]

((c*d^2 - a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(d + e*x)^3*(15*c^4*d^4 - (15*e^4
*(a*e + c*d*x)^4)/(d + e*x)^4 + (70*c*d*e^3*(a*e + c*d*x)^3)/(d + e*x)^3 + (128*c^2*d^2*e^2*(a*e + c*d*x)^2)/(
d + e*x)^2 - (70*c^3*d^3*e*(a*e + c*d*x))/(d + e*x)))/((c*d^2 - a*e^2)^5*(a*e + c*d*x)) - (15*ArcTanh[(Sqrt[e]
*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(640*c^(5/2)*d^(
5/2)*e^(7/2))

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Maple [A]
time = 0.09, size = 327, normalized size = 1.19

method	result	size
default	\(\frac {\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c d e}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2}}{e}\)	\(327\)

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),x,method=_RETURNVERBOSE)

[Out]

1/e*(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)+a*e^2-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)+a*e^2-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*a*e^2-1/2*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. 896 vs. \(2 (241) = 482\).
time = 0.35, size = 896, normalized size = 3.27 \begin {gather*} -\frac {3 \, c^{4} d^{9} 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 )}{256 \, \left (c d\right )^{\frac {3}{2}}} + \frac {15 \, a c^{3} d^{7} 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 )}{256 \, \left (c d\right )^{\frac {3}{2}}} + \frac {3}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} c^{2} d^{5} x e^{\left (-2\right )} + \frac {3}{128} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} c^{2} d^{6} e^{\left (-3\right )} - \frac {15 \, a^{2} c^{2} d^{5} 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 )}{128 \, \left (c d\right )^{\frac {3}{2}}} - \frac {3}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a c d^{4} e^{\left (-1\right )} + \frac {15 \, a^{3} c d^{3} 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 )}{128 \, \left (c d\right )^{\frac {3}{2}}} - \frac {9}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a c d^{3} x - \frac {1}{8} \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}} c d^{2} x e^{\left (-1\right )} - \frac {1}{16} \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}} c d^{3} e^{\left (-2\right )} + \frac {9}{64} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{2} d x e^{2} - \frac {15 \, a^{4} d 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 )}{256 \, \left (c d\right )^{\frac {3}{2}}} + \frac {1}{8} \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}} a x e - \frac {3 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{3} x e^{4}}{64 \, c d} + \frac {3 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{3} e^{3}}{64 \, c} + \frac {1}{5} \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {5}{2}} e^{\left (-1\right )} + \frac {3 \, a^{5} e^{\frac {13}{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 )}{256 \, \left (c d\right )^{\frac {3}{2}} c d} + \frac {{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}} a^{2} e^{2}}{16 \, c d} - \frac {3 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{4} e^{5}}{128 \, c^{2} d^{2}} \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),x, algorithm="maxima")

[Out]

-3/256*c^4*d^9*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) + 15/256*a*c^3*d^7*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) + 3/64*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*c
^2*d^5*x*e^(-2) + 3/128*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*c^2*d^6*e^(-3) - 15/128*a^2*c^2*d^5*e^(1/2
)*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) - 3/64*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*a*c*d^4*e^(-1) + 15/128*a^3*c*d^3*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) - 9/64*sq
rt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*a*c*d^3*x - 1/8*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^(3/2)*c*d^2*
x*e^(-1) - 1/16*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^(3/2)*c*d^3*e^(-2) + 9/64*sqrt(c*d*x^2*e + c*d^2*x + a
*x*e^2 + a*d*e)*a^2*d*x*e^2 - 15/256*a^4*d*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) + 1/8*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^(3/2)*a*x*
e - 3/64*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*a^3*x*e^4/(c*d) + 3/64*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e)*a^3*e^3/c + 1/5*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^(5/2)*e^(-1) + 3/256*a^5*e^(13/2)*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)*c*d) +
1/16*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^(3/2)*a^2*e^2/(c*d) - 3/128*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e)*a^4*e^5/(c^2*d^2)

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Fricas [A]
time = 2.76, size = 815, normalized size = 2.97 \begin {gather*} \left [\frac {{\left (15 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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^{5} d^{8} x e^{2} - 15 \, c^{5} d^{9} e - 10 \, a^{3} c^{2} d^{2} x e^{8} + 15 \, a^{4} c d e^{9} - 2 \, {\left (124 \, a^{2} c^{3} d^{3} x^{2} + 35 \, a^{3} c^{2} d^{3}\right )} e^{7} - 2 \, {\left (168 \, a c^{4} d^{4} x^{3} + 233 \, a^{2} c^{3} d^{4} x\right )} e^{6} - 128 \, {\left (c^{5} d^{5} x^{4} + 4 \, a c^{4} d^{5} x^{2} + a^{2} c^{3} d^{5}\right )} e^{5} - 2 \, {\left (88 \, c^{5} d^{6} x^{3} + 23 \, a c^{4} d^{6} x\right )} e^{4} - 2 \, {\left (4 \, c^{5} d^{7} x^{2} - 35 \, a c^{4} d^{7}\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 )}}{2560 \, c^{3} d^{3}}, \frac {{\left (15 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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^{5} d^{8} x e^{2} - 15 \, c^{5} d^{9} e - 10 \, a^{3} c^{2} d^{2} x e^{8} + 15 \, a^{4} c d e^{9} - 2 \, {\left (124 \, a^{2} c^{3} d^{3} x^{2} + 35 \, a^{3} c^{2} d^{3}\right )} e^{7} - 2 \, {\left (168 \, a c^{4} d^{4} x^{3} + 233 \, a^{2} c^{3} d^{4} x\right )} e^{6} - 128 \, {\left (c^{5} d^{5} x^{4} + 4 \, a c^{4} d^{5} x^{2} + a^{2} c^{3} d^{5}\right )} e^{5} - 2 \, {\left (88 \, c^{5} d^{6} x^{3} + 23 \, a c^{4} d^{6} x\right )} e^{4} - 2 \, {\left (4 \, c^{5} d^{7} x^{2} - 35 \, a c^{4} d^{7}\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 )}}{1280 \, c^{3} d^{3}}\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),x, algorithm="fricas")

[Out]

[1/2560*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10
)*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^5*d^8
*x*e^2 - 15*c^5*d^9*e - 10*a^3*c^2*d^2*x*e^8 + 15*a^4*c*d*e^9 - 2*(124*a^2*c^3*d^3*x^2 + 35*a^3*c^2*d^3)*e^7 -
2*(168*a*c^4*d^4*x^3 + 233*a^2*c^3*d^4*x)*e^6 - 128*(c^5*d^5*x^4 + 4*a*c^4*d^5*x^2 + a^2*c^3*d^5)*e^5 - 2*(88
*c^5*d^6*x^3 + 23*a*c^4*d^6*x)*e^4 - 2*(4*c^5*d^7*x^2 - 35*a*c^4*d^7)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 +
a*d)*e))*e^(-4)/(c^3*d^3), 1/1280*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 +
5*a^4*c*d^2*e^8 - a^5*e^10)*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^5*d^8*x*e^2 -
15*c^5*d^9*e - 10*a^3*c^2*d^2*x*e^8 + 15*a^4*c*d*e^9 - 2*(124*a^2*c^3*d^3*x^2 + 35*a^3*c^2*d^3)*e^7 - 2*(168*
a*c^4*d^4*x^3 + 233*a^2*c^3*d^4*x)*e^6 - 128*(c^5*d^5*x^4 + 4*a*c^4*d^5*x^2 + a^2*c^3*d^5)*e^5 - 2*(88*c^5*d^6
*x^3 + 23*a*c^4*d^6*x)*e^4 - 2*(4*c^5*d^7*x^2 - 35*a*c^4*d^7)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)
)*e^(-4)/(c^3*d^3)]

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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),x)

[Out]

Timed out

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Giac [A]
time = 1.75, size = 396, normalized size = 1.45 \begin {gather*} \frac {1}{640} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, c^{2} d^{2} x e + \frac {{\left (11 \, c^{6} d^{7} e^{4} + 21 \, a c^{5} d^{5} e^{6}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (c^{6} d^{8} e^{3} + 64 \, a c^{5} d^{6} e^{5} + 31 \, a^{2} c^{4} d^{4} e^{7}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x - \frac {{\left (5 \, c^{6} d^{9} e^{2} - 23 \, a c^{5} d^{7} e^{4} - 233 \, a^{2} c^{4} d^{5} e^{6} - 5 \, a^{3} c^{3} d^{3} e^{8}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (15 \, c^{6} d^{10} e - 70 \, a c^{5} d^{8} e^{3} + 128 \, a^{2} c^{4} d^{6} e^{5} + 70 \, a^{3} c^{3} d^{4} e^{7} - 15 \, a^{4} c^{2} d^{2} e^{9}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} + \frac {3 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{256 \, \sqrt {c d} c^{2} d^{2}} \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),x, algorithm="giac")

[Out]

1/640*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*c^2*d^2*x*e + (11*c^6*d^7*e^4 + 21*a*c^5*d^5*e^6
)*e^(-4)/(c^4*d^4))*x + (c^6*d^8*e^3 + 64*a*c^5*d^6*e^5 + 31*a^2*c^4*d^4*e^7)*e^(-4)/(c^4*d^4))*x - (5*c^6*d^9
*e^2 - 23*a*c^5*d^7*e^4 - 233*a^2*c^4*d^5*e^6 - 5*a^3*c^3*d^3*e^8)*e^(-4)/(c^4*d^4))*x + (15*c^6*d^10*e - 70*a
*c^5*d^8*e^3 + 128*a^2*c^4*d^6*e^5 + 70*a^3*c^3*d^4*e^7 - 15*a^4*c^2*d^2*e^9)*e^(-4)/(c^4*d^4)) + 3/256*(c^5*d
^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*e^(-7/2)*log(abs
(-c*d^2 - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*sqrt(c*d)*e^(1/2) - a*e^2))/(s
qrt(c*d)*c^2*d^2)

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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}}{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)^(5/2)/(d + e*x),x)

[Out]

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

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