3.5.66 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^6 (d+e x)} \, dx\) [466]

Optimal. Leaf size=289 \[ \frac {3 \left (c d^2-a e^2\right )^3 \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac {\left (\frac {c}{a e}-\frac {e}{d^2}\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac {3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 a^{5/2} d^{7/2} e^{5/2}} \]

[Out]

-1/16*(c/a/e-e/d^2)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^4-1/5*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(5/2)/d/x^5-3/256*(-a*e^2+c*d^2)^5*arctanh(1/2*(2*a*d*e+(a*e^2+c*d^2)*x)/a^(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))/a^(5/2)/d^(7/2)/e^(5/2)+3/128*(-a*e^2+c*d^2)^3*(2*a*d*e+(a*e^2+c
*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d^3/e^2/x^2

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Rubi [A]
time = 0.21, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {863, 820, 734, 738, 212} \begin {gather*} -\frac {3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 a^{5/2} d^{7/2} e^{5/2}}+\frac {3 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 d x^5}-\frac {\left (\frac {c}{a e}-\frac {e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{16 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)/(x^6*(d + e*x)),x]

[Out]

(3*(c*d^2 - a*e^2)^3*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*a^2*d^3*e
^2*x^2) - ((c/(a*e) - e/d^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(16*
x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*d*x^5) - (3*(c*d^2 - a*e^2)^5*ArcTanh[(2*a*d*e + (c*d^
2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(256*a^(5/2)*d^(7/2)*e
^(5/2))

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 734

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

Rule 738

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

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 863

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

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}}{x^6 (d+e x)} \, dx &=\int \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}}{x^6} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac {\left (-2 a c d^2 e+a e \left (c d^2+a e^2\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5} \, dx}{2 a d e}\\ &=-\frac {\left (\frac {c}{a e}-\frac {e}{d^2}\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac {\left (3 \left (c d^2-a e^2\right )^3\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{32 a d^2 e}\\ &=\frac {3 \left (c d^2-a e^2\right )^3 \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac {\left (\frac {c}{a e}-\frac {e}{d^2}\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}+\frac {\left (3 \left (c d^2-a e^2\right )^5\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 a^2 d^3 e^2}\\ &=\frac {3 \left (c d^2-a e^2\right )^3 \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac {\left (\frac {c}{a e}-\frac {e}{d^2}\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac {\left (3 \left (c d^2-a e^2\right )^5\right ) \text {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 a^2 d^3 e^2}\\ &=\frac {3 \left (c d^2-a e^2\right )^3 \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac {\left (\frac {c}{a e}-\frac {e}{d^2}\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac {3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 a^{5/2} d^{7/2} e^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 271, normalized size = 0.94 \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 {a} \sqrt {d} \sqrt {e} (d+e x)^3 \left (15 a^4 e^4-\frac {15 d^4 (a e+c d x)^4}{(d+e x)^4}+\frac {70 a d^3 e (a e+c d x)^3}{(d+e x)^3}+\frac {128 a^2 d^2 e^2 (a e+c d x)^2}{(d+e x)^2}-\frac {70 a^3 d e^3 (a e+c d x)}{d+e x}\right )}{\left (-c d^2+a e^2\right )^5 x^5 (a e+c d x)}+\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{640 a^{5/2} d^{7/2} e^{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)/(x^6*(d + e*x)),x]

[Out]

((-(c*d^2) + a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(d + e*x)^3*(15*a^4*e^4 - (1
5*d^4*(a*e + c*d*x)^4)/(d + e*x)^4 + (70*a*d^3*e*(a*e + c*d*x)^3)/(d + e*x)^3 + (128*a^2*d^2*e^2*(a*e + c*d*x)
^2)/(d + e*x)^2 - (70*a^3*d*e^3*(a*e + c*d*x))/(d + e*x)))/((-(c*d^2) + a*e^2)^5*x^5*(a*e + c*d*x))) + (15*Arc
Tanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(64
0*a^(5/2)*d^(7/2)*e^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(19538\) vs. \(2(259)=518\).
time = 0.08, size = 19539, normalized size = 67.61

method	result	size
default	\(\text {Expression too large to display}\)	\(19539\)

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)/x^6/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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)/x^6/(e*x+d),x, algorithm="maxima")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((x*e + d)*x^6), x)

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Fricas [A]
time = 21.73, size = 901, normalized size = 3.12 \begin {gather*} \left [\frac {{\left (15 \, {\left (c^{5} d^{10} x^{5} - 5 \, a c^{4} d^{8} x^{5} e^{2} + 10 \, a^{2} c^{3} d^{6} x^{5} e^{4} - 10 \, a^{3} c^{2} d^{4} x^{5} e^{6} + 5 \, a^{4} c d^{2} x^{5} e^{8} - a^{5} x^{5} e^{10}\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} - 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) + 4 \, {\left (15 \, a c^{4} d^{9} x^{4} e - 10 \, a^{2} c^{3} d^{8} x^{3} e^{2} - 15 \, a^{5} d x^{4} e^{9} + 10 \, a^{5} d^{2} x^{3} e^{8} + 2 \, {\left (35 \, a^{4} c d^{3} x^{4} - 4 \, a^{5} d^{3} x^{2}\right )} e^{7} - 2 \, {\left (23 \, a^{4} c d^{4} x^{3} + 88 \, a^{5} d^{4} x\right )} e^{6} - 128 \, {\left (a^{3} c^{2} d^{5} x^{4} + 4 \, a^{4} c d^{5} x^{2} + a^{5} d^{5}\right )} e^{5} - 2 \, {\left (233 \, a^{3} c^{2} d^{6} x^{3} + 168 \, a^{4} c d^{6} x\right )} e^{4} - 2 \, {\left (35 \, a^{2} c^{3} d^{7} x^{4} + 124 \, a^{3} c^{2} d^{7} x^{2}\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 (-3\right )}}{2560 \, a^{3} d^{4} x^{5}}, \frac {{\left (15 \, {\left (c^{5} d^{10} x^{5} - 5 \, a c^{4} d^{8} x^{5} e^{2} + 10 \, a^{2} c^{3} d^{6} x^{5} e^{4} - 10 \, a^{3} c^{2} d^{4} x^{5} e^{6} + 5 \, a^{4} c d^{2} x^{5} e^{8} - a^{5} x^{5} e^{10}\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (15 \, a c^{4} d^{9} x^{4} e - 10 \, a^{2} c^{3} d^{8} x^{3} e^{2} - 15 \, a^{5} d x^{4} e^{9} + 10 \, a^{5} d^{2} x^{3} e^{8} + 2 \, {\left (35 \, a^{4} c d^{3} x^{4} - 4 \, a^{5} d^{3} x^{2}\right )} e^{7} - 2 \, {\left (23 \, a^{4} c d^{4} x^{3} + 88 \, a^{5} d^{4} x\right )} e^{6} - 128 \, {\left (a^{3} c^{2} d^{5} x^{4} + 4 \, a^{4} c d^{5} x^{2} + a^{5} d^{5}\right )} e^{5} - 2 \, {\left (233 \, a^{3} c^{2} d^{6} x^{3} + 168 \, a^{4} c d^{6} x\right )} e^{4} - 2 \, {\left (35 \, a^{2} c^{3} d^{7} x^{4} + 124 \, a^{3} c^{2} d^{7} x^{2}\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 (-3\right )}}{1280 \, a^{3} d^{4} x^{5}}\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)/x^6/(e*x+d),x, algorithm="fricas")

[Out]

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

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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)/x**6/(e*x+d),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2480 vs. \(2 (258) = 516\).
time = 1.55, size = 2480, normalized size = 8.58 \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)^(5/2)/x^6/(e*x+d),x, algorithm="giac")

[Out]

3/128*(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)*arct
an(-(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(-a*d*e))*e^(-2)/(sqrt(-a*d*e)*a^2
*d^3) - 1/640*(15*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^4*c^5*d^14*e^4 - 70*(s
qrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^3*c^5*d^13*e^3 + 128*(sqrt(c*d)*x*e^(1/2
) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^2*c^5*d^12*e^2 + 70*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e
+ c*d^2*x + a*x*e^2 + a*d*e))^7*a*c^5*d^11*e - 15*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e))^9*c^5*d^10 + 1280*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^6*sqrt(c*d)*a^2
*c^4*d^10*e^(5/2) - 75*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^5*c^4*d^12*e^6 +
350*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^4*c^4*d^11*e^5 + 3200*(sqrt(c*d)*x
*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^3*c^4*d^10*e^4 + 2210*(sqrt(c*d)*x*e^(1/2) - sqrt(
c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^2*c^4*d^9*e^3 + 75*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x
+ a*x*e^2 + a*d*e))^9*a*c^4*d^8*e^2 + 6400*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)
)^4*sqrt(c*d)*a^4*c^3*d^9*e^(11/2) + 6400*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^
6*sqrt(c*d)*a^3*c^3*d^8*e^(9/2) + 2560*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^8*s
qrt(c*d)*a^2*c^3*d^7*e^(7/2) + 150*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^6*c^3
*d^10*e^8 + 5700*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^5*c^3*d^9*e^7 + 12800
*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^4*c^3*d^8*e^6 + 7100*(sqrt(c*d)*x*e^(
1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^3*c^3*d^7*e^5 + 1130*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x
^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^2*c^3*d^6*e^4 + 256*sqrt(c*d)*a^7*c^2*d^9*e^(19/2) + 2560*(sqrt(c*d)*x*
e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*sqrt(c*d)*a^6*c^2*d^8*e^(17/2) + 14080*(sqrt(c*d)*x*e
^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^4*sqrt(c*d)*a^5*c^2*d^7*e^(15/2) + 11520*(sqrt(c*d)*x*e^
(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^6*sqrt(c*d)*a^4*c^2*d^6*e^(13/2) + 3840*(sqrt(c*d)*x*e^(1
/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^8*sqrt(c*d)*a^3*c^2*d^5*e^(11/2) + 1130*(sqrt(c*d)*x*e^(1/2
) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^7*c^2*d^8*e^10 + 7100*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e
+ c*d^2*x + a*x*e^2 + a*d*e))^3*a^6*c^2*d^7*e^9 + 12800*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x
*e^2 + a*d*e))^5*a^5*c^2*d^6*e^8 + 5700*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*
a^4*c^2*d^5*e^7 + 150*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^3*c^2*d^4*e^6 +
2560*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*sqrt(c*d)*a^7*c*d^6*e^(21/2) + 6400
*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^4*sqrt(c*d)*a^6*c*d^5*e^(19/2) + 6400*(sq
rt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^6*sqrt(c*d)*a^5*c*d^4*e^(17/2) + 75*(sqrt(c*d
)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^8*c*d^6*e^12 + 2210*(sqrt(c*d)*x*e^(1/2) - sqrt(c
*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^7*c*d^5*e^11 + 3200*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x
+ a*x*e^2 + a*d*e))^5*a^6*c*d^4*e^10 + 350*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)
)^7*a^5*c*d^3*e^9 - 75*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^4*c*d^2*e^8 + 1
280*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^4*sqrt(c*d)*a^7*d^3*e^(23/2) - 15*(sqr
t(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^9*d^4*e^14 + 70*(sqrt(c*d)*x*e^(1/2) - sqrt(
c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^8*d^3*e^13 + 128*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x +
a*x*e^2 + a*d*e))^5*a^7*d^2*e^12 - 70*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a
^6*d*e^11 + 15*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^5*e^10)*e^(-2)/((a*d*e
- (sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2)^5*a^2*d^3)

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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}}{x^6\,\left (d+e\,x\right )} \,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)/(x^6*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)), x)

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