\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\) [2051]

Optimal result

Integrand size = 39, antiderivative size = 240 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt {d+e x}}+\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {2 \left (c d^2-a e^2\right )^{5/2} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{e^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {678, 674, 211} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (c d^2-a e^2\right )^{5/2} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{e^{7/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}+\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 \sqrt {d+e x}} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(7/2),x]

[Out]

(2*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^3*Sqrt[d + e*x]) + (2*(a - (c*d^2)/e^2)*(
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)
^(5/2))/(5*e*(d + e*x)^(5/2)) - (2*(c*d^2 - a*e^2)^(5/2)*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/e^(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 674

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

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*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {\left (2 c d^2 e-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}}{(d+e x)^{5/2}} \, dx}{e^2} \\ & = \frac {2 \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}}{3 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {\left (c d^2-a e^2\right )^2 \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{3/2}} \, dx}{e^2} \\ & = \frac {2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt {d+e x}}+\frac {2 \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}}{3 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {\left (c d^2-a e^2\right )^3 \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{e^3} \\ & = \frac {2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt {d+e x}}+\frac {2 \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}}{3 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {\left (2 \left (c d^2-a e^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 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 )}{e^2} \\ & = \frac {2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt {d+e x}}+\frac {2 \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}}{3 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {2 \left (c d^2-a e^2\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {e} \sqrt {a e+c d x} \left (23 a^2 e^4+a c d e^2 (-35 d+11 e x)+c^2 d^2 \left (15 d^2-5 d e x+3 e^2 x^2\right )\right )-15 \left (c d^2-a e^2\right )^{5/2} \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{15 e^{7/2} \sqrt {(a e+c d x) (d+e x)}} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(7/2),x]

[Out]

(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(23*a^2*e^4 + a*c*d*e^2*(-35*d + 11*e*x) + c^2*d
^2*(15*d^2 - 5*d*e*x + 3*e^2*x^2)) - 15*(c*d^2 - a*e^2)^(5/2)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 -
a*e^2]]))/(15*e^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(426\) vs. \(2(212)=424\).

Time = 2.95 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.78

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a^{3} e^{6}-45 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a^{2} c \,d^{2} e^{4}+45 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{4} e^{2}-15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{6}-3 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-11 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+5 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-23 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}+35 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}-15 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, e^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(427\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/15*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a^3*e^6-45*arctanh(
e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a^2*c*d^2*e^4+45*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1
/2))*a*c^2*d^4*e^2-15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^6-3*c^2*d^2*e^2*x^2*(c*d*x+a*
e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-11*a*c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+5*c^2*d^3*e*x*(c*d*x
+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-23*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a^2*e^4+35*(c*d*x+a*e)^(1/2)*
((a*e^2-c*d^2)*e)^(1/2)*a*c*d^2*e^2-15*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^4)/(e*x+d)^(1/2)/(c*d*x
+a*e)^(1/2)/e^3/((a*e^2-c*d^2)*e)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 538, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\left [\frac {15 \, {\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{e}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} e \sqrt {-\frac {c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 35 \, a c d^{2} e^{2} + 23 \, a^{2} e^{4} - {\left (5 \, c^{2} d^{3} e - 11 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{15 \, {\left (e^{4} x + d e^{3}\right )}}, \frac {2 \, {\left (15 \, {\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt {\frac {c d^{2} - a e^{2}}{e}} \arctan \left (\frac {\sqrt {e x + d} \sqrt {\frac {c d^{2} - a e^{2}}{e}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\right ) + {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 35 \, a c d^{2} e^{2} + 23 \, a^{2} e^{4} - {\left (5 \, c^{2} d^{3} e - 11 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}\right )}}{15 \, {\left (e^{4} x + d e^{3}\right )}}\right ] \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

[1/15*(15*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(-(c*d^2 - a*e^2
)/e)*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*
x + d)*e*sqrt(-(c*d^2 - a*e^2)/e))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(3*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 35*a*c*d^2
*e^2 + 23*a^2*e^4 - (5*c^2*d^3*e - 11*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))
/(e^4*x + d*e^3), 2/15*(15*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqr
t((c*d^2 - a*e^2)/e)*arctan(sqrt(e*x + d)*sqrt((c*d^2 - a*e^2)/e)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))
+ (3*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 35*a*c*d^2*e^2 + 23*a^2*e^4 - (5*c^2*d^3*e - 11*a*c*d*e^3)*x)*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(e^4*x + d*e^3)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^(7/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (212) = 424\).

Time = 0.32 (sec) , antiderivative size = 644, normalized size of antiderivative = 2.68 \[ \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)^{7/2}} \, dx=-\frac {2 \, {\left (\frac {15 \, {\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}}} - \frac {15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{4} e^{14} {\left | e \right |} - 30 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c d^{2} e^{16} {\left | e \right |} + 15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} e^{18} {\left | e \right |} - 5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d^{2} e^{13} {\left | e \right |} + 5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{15} {\left | e \right |} + 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} e^{12} {\left | e \right |}}{e^{15}}\right )}}{15 \, e^{4}} + \frac {2 \, {\left (15 \, c^{3} d^{6} e {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) - 45 \, a c^{2} d^{4} e^{3} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) + 45 \, a^{2} c d^{2} e^{5} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) - 15 \, a^{3} e^{7} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) - 23 \, \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} {\left | e \right |} + 46 \, \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} {\left | e \right |} - 23 \, \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4} {\left | e \right |}\right )}}{15 \, \sqrt {c d^{2} e - a e^{3}} e^{5}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

-2/15*(15*(c^3*d^6*abs(e) - 3*a*c^2*d^4*e^2*abs(e) + 3*a^2*c*d^2*e^4*abs(e) - a^3*e^6*abs(e))*arctan(sqrt((e*x
+ d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))/sqrt(c*d^2*e - a*e^3) - (15*sqrt((e*x + d)*c*d*e - c*d^2
*e + a*e^3)*c^2*d^4*e^14*abs(e) - 30*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c*d^2*e^16*abs(e) + 15*sqrt((e*
x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*e^18*abs(e) - 5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c*d^2*e^13*abs(e
) + 5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^15*abs(e) + 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*e^
12*abs(e))/e^15)/e^4 + 2/15*(15*c^3*d^6*e*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3)) - 45*a*c
^2*d^4*e^3*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3)) + 45*a^2*c*d^2*e^5*abs(e)*arctan(sqrt(-
c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3)) - 15*a^3*e^7*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3
)) - 23*sqrt(c*d^2*e - a*e^3)*sqrt(-c*d^2*e + a*e^3)*c^2*d^4*abs(e) + 46*sqrt(c*d^2*e - a*e^3)*sqrt(-c*d^2*e +
a*e^3)*a*c*d^2*e^2*abs(e) - 23*sqrt(c*d^2*e - a*e^3)*sqrt(-c*d^2*e + a*e^3)*a^2*e^4*abs(e))/(sqrt(c*d^2*e - a
*e^3)*e^5)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(7/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(7/2), x)