\(\int \frac {(d+e x)^{13/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [2019]

Optimal result

Integrand size = 37, antiderivative size = 186 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{4 c^4 d^4}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}-\frac {35 e^2 \left (c d^2-a e^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}} \]

[Out]

35/12*e^2*(e*x+d)^(3/2)/c^3/d^3-7/4*e*(e*x+d)^(5/2)/c^2/d^2/(c*d*x+a*e)-1/2*(e*x+d)^(7/2)/c/d/(c*d*x+a*e)^2-35
/4*e^2*(-a*e^2+c*d^2)^(3/2)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2))/c^(9/2)/d^(9/2)+35/4*e
^2*(-a*e^2+c*d^2)*(e*x+d)^(1/2)/c^4/d^4

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {35 e^2 \left (c d^2-a e^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}}+\frac {35 e^2 \sqrt {d+e x} \left (c d^2-a e^2\right )}{4 c^4 d^4}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3} \]

[In]

Int[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(35*e^2*(c*d^2 - a*e^2)*Sqrt[d + e*x])/(4*c^4*d^4) + (35*e^2*(d + e*x)^(3/2))/(12*c^3*d^3) - (7*e*(d + e*x)^(5
/2))/(4*c^2*d^2*(a*e + c*d*x)) - (d + e*x)^(7/2)/(2*c*d*(a*e + c*d*x)^2) - (35*e^2*(c*d^2 - a*e^2)^(3/2)*ArcTa
nh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(4*c^(9/2)*d^(9/2))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 640

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^{7/2}}{(a e+c d x)^3} \, dx \\ & = -\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{(a e+c d x)^2} \, dx}{4 c d} \\ & = -\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {\left (35 e^2\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{8 c^2 d^2} \\ & = \frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{8 c^3 d^3} \\ & = \frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{4 c^4 d^4}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{8 c^4 d^4} \\ & = \frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{4 c^4 d^4}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {\left (35 e \left (c d^2-a e^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 c^4 d^4} \\ & = \frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{4 c^4 d^4}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}-\frac {35 e^2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (105 a^3 e^6-35 a^2 c d e^4 (4 d-5 e x)+7 a c^2 d^2 e^2 \left (3 d^2-34 d e x+8 e^2 x^2\right )+c^3 d^3 \left (6 d^3+39 d^2 e x-80 d e^2 x^2-8 e^3 x^3\right )\right )}{12 c^4 d^4 (a e+c d x)^2}+\frac {35 \left (c d^2 e-a e^3\right )^2 \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{4 c^{9/2} d^{9/2} \sqrt {-c d^2+a e^2}} \]

[In]

Integrate[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-1/12*(Sqrt[d + e*x]*(105*a^3*e^6 - 35*a^2*c*d*e^4*(4*d - 5*e*x) + 7*a*c^2*d^2*e^2*(3*d^2 - 34*d*e*x + 8*e^2*x
^2) + c^3*d^3*(6*d^3 + 39*d^2*e*x - 80*d*e^2*x^2 - 8*e^3*x^3)))/(c^4*d^4*(a*e + c*d*x)^2) + (35*(c*d^2*e - a*e
^3)^2*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(4*c^(9/2)*d^(9/2)*Sqrt[-(c*d^2) + a*e^2
])

Maple [A] (verified)

Time = 9.71 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {2 e^{2} \left (-x c d e +9 e^{2} a -10 c \,d^{2}\right ) \sqrt {e x +d}}{3 c^{4} d^{4}}+\frac {\left (2 a^{2} e^{4}-4 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) e^{2} \left (\frac {-\frac {13 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (-\frac {11 e^{2} a}{8}+\frac {11 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {35 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{4} d^{4}}\)	\(180\)
pseudoelliptic	\(\frac {\frac {35 e^{2} \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right )^{2} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{4}-\frac {35 \left (\frac {2 \left (-\frac {4}{3} e^{3} x^{3}-\frac {40}{3} d \,e^{2} x^{2}+\frac {13}{2} d^{2} e x +d^{3}\right ) d^{3} c^{3}}{35}+\frac {e^{2} \left (\frac {8}{3} x^{2} e^{2}-\frac {34}{3} d e x +d^{2}\right ) d^{2} a \,c^{2}}{5}-\frac {4 e^{4} \left (-\frac {5 e x}{4}+d \right ) d \,a^{2} c}{3}+e^{6} a^{3}\right ) \sqrt {e x +d}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}{4}}{c^{4} d^{4} \left (c d x +a e \right )^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\)	\(208\)
derivativedivides	\(2 e^{2} \left (-\frac {-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a \,e^{2} \sqrt {e x +d}-3 c \,d^{2} \sqrt {e x +d}}{c^{4} d^{4}}+\frac {\frac {\left (-\frac {13}{8} d \,e^{4} a^{2} c +\frac {13}{4} d^{3} e^{2} c^{2} a -\frac {13}{8} d^{5} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} e^{6} a^{3}+\frac {33}{8} d^{2} e^{4} a^{2} c -\frac {33}{8} d^{4} e^{2} c^{2} a +\frac {11}{8} c^{3} d^{6}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {35 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{4} d^{4}}\right )\)	\(245\)
default	\(2 e^{2} \left (-\frac {-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a \,e^{2} \sqrt {e x +d}-3 c \,d^{2} \sqrt {e x +d}}{c^{4} d^{4}}+\frac {\frac {\left (-\frac {13}{8} d \,e^{4} a^{2} c +\frac {13}{4} d^{3} e^{2} c^{2} a -\frac {13}{8} d^{5} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} e^{6} a^{3}+\frac {33}{8} d^{2} e^{4} a^{2} c -\frac {33}{8} d^{4} e^{2} c^{2} a +\frac {11}{8} c^{3} d^{6}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {35 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{4} d^{4}}\right )\)	\(245\)

[In]

int((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

-2/3*e^2*(-c*d*e*x+9*a*e^2-10*c*d^2)*(e*x+d)^(1/2)/c^4/d^4+1/c^4/d^4*(2*a^2*e^4-4*a*c*d^2*e^2+2*c^2*d^4)*e^2*(
(-13/8*c*d*(e*x+d)^(3/2)+(-11/8*e^2*a+11/8*c*d^2)*(e*x+d)^(1/2))/(c*d*(e*x+d)+e^2*a-c*d^2)^2+35/8/((a*e^2-c*d^
2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (154) = 308\).

Time = 0.34 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.43 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\left [\frac {105 \, {\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{3} - 6 \, c^{3} d^{6} - 21 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 8 \, {\left (10 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} - {\left (39 \, c^{3} d^{5} e - 238 \, a c^{2} d^{3} e^{3} + 175 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}, -\frac {105 \, {\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (8 \, c^{3} d^{3} e^{3} x^{3} - 6 \, c^{3} d^{6} - 21 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 8 \, {\left (10 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} - {\left (39 \, c^{3} d^{5} e - 238 \, a c^{2} d^{3} e^{3} + 175 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{12 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}\right ] \]

[In]

integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

[1/24*(105*(a^2*c*d^2*e^4 - a^3*e^6 + (c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 2*(a*c^2*d^3*e^3 - a^2*c*d*e^5)*x)*s
qrt((c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*sqrt(e*x + d)*c*d*sqrt((c*d^2 - a*e^2)/(c*d)))/(
c*d*x + a*e)) + 2*(8*c^3*d^3*e^3*x^3 - 6*c^3*d^6 - 21*a*c^2*d^4*e^2 + 140*a^2*c*d^2*e^4 - 105*a^3*e^6 + 8*(10*
c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 - (39*c^3*d^5*e - 238*a*c^2*d^3*e^3 + 175*a^2*c*d*e^5)*x)*sqrt(e*x + d))/(c
^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d^4*e^2), -1/12*(105*(a^2*c*d^2*e^4 - a^3*e^6 + (c^3*d^4*e^2 - a*c^2*d^
2*e^4)*x^2 + 2*(a*c^2*d^3*e^3 - a^2*c*d*e^5)*x)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(-sqrt(e*x + d)*c*d*sqrt(-(
c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e^2)) - (8*c^3*d^3*e^3*x^3 - 6*c^3*d^6 - 21*a*c^2*d^4*e^2 + 140*a^2*c*d^2*e^4
- 105*a^3*e^6 + 8*(10*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 - (39*c^3*d^5*e - 238*a*c^2*d^3*e^3 + 175*a^2*c*d*e^
5)*x)*sqrt(e*x + d))/(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d^4*e^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(13/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {35 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{4 \, \sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} - \frac {13 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{5} e^{2} - 11 \, \sqrt {e x + d} c^{3} d^{6} e^{2} - 26 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{4} + 33 \, \sqrt {e x + d} a c^{2} d^{4} e^{4} + 13 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} c d e^{6} - 33 \, \sqrt {e x + d} a^{2} c d^{2} e^{6} + 11 \, \sqrt {e x + d} a^{3} e^{8}}{4 \, {\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}^{2} c^{4} d^{4}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{6} d^{6} e^{2} + 9 \, \sqrt {e x + d} c^{6} d^{7} e^{2} - 9 \, \sqrt {e x + d} a c^{5} d^{5} e^{4}\right )}}{3 \, c^{9} d^{9}} \]

[In]

integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

35/4*(c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*arctan(sqrt(e*x + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))/(sqrt(-c^2*d
^3 + a*c*d*e^2)*c^4*d^4) - 1/4*(13*(e*x + d)^(3/2)*c^3*d^5*e^2 - 11*sqrt(e*x + d)*c^3*d^6*e^2 - 26*(e*x + d)^(
3/2)*a*c^2*d^3*e^4 + 33*sqrt(e*x + d)*a*c^2*d^4*e^4 + 13*(e*x + d)^(3/2)*a^2*c*d*e^6 - 33*sqrt(e*x + d)*a^2*c*
d^2*e^6 + 11*sqrt(e*x + d)*a^3*e^8)/(((e*x + d)*c*d - c*d^2 + a*e^2)^2*c^4*d^4) + 2/3*((e*x + d)^(3/2)*c^6*d^6
*e^2 + 9*sqrt(e*x + d)*c^6*d^7*e^2 - 9*sqrt(e*x + d)*a*c^5*d^5*e^4)/(c^9*d^9)

Mupad [B] (verification not implemented)

Time = 10.00 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.72 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {2\,e^2\,{\left (d+e\,x\right )}^{3/2}}{3\,c^3\,d^3}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (\frac {13\,a^2\,c\,d\,e^6}{4}-\frac {13\,a\,c^2\,d^3\,e^4}{2}+\frac {13\,c^3\,d^5\,e^2}{4}\right )+\sqrt {d+e\,x}\,\left (\frac {11\,a^3\,e^8}{4}-\frac {33\,a^2\,c\,d^2\,e^6}{4}+\frac {33\,a\,c^2\,d^4\,e^4}{4}-\frac {11\,c^3\,d^6\,e^2}{4}\right )}{c^6\,d^8-\left (2\,c^6\,d^7-2\,a\,c^5\,d^5\,e^2\right )\,\left (d+e\,x\right )+c^6\,d^6\,{\left (d+e\,x\right )}^2-2\,a\,c^5\,d^6\,e^2+a^2\,c^4\,d^4\,e^4}+\frac {2\,e^2\,\left (3\,c^3\,d^4-3\,a\,c^2\,d^2\,e^2\right )\,\sqrt {d+e\,x}}{c^6\,d^6}+\frac {35\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e^2\,{\left (a\,e^2-c\,d^2\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^6-2\,a\,c\,d^2\,e^4+c^2\,d^4\,e^2}\right )\,{\left (a\,e^2-c\,d^2\right )}^{3/2}}{4\,c^{9/2}\,d^{9/2}} \]

[In]

int((d + e*x)^(13/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

(2*e^2*(d + e*x)^(3/2))/(3*c^3*d^3) - ((d + e*x)^(3/2)*((13*c^3*d^5*e^2)/4 - (13*a*c^2*d^3*e^4)/2 + (13*a^2*c*
d*e^6)/4) + (d + e*x)^(1/2)*((11*a^3*e^8)/4 - (11*c^3*d^6*e^2)/4 + (33*a*c^2*d^4*e^4)/4 - (33*a^2*c*d^2*e^6)/4
))/(c^6*d^8 - (2*c^6*d^7 - 2*a*c^5*d^5*e^2)*(d + e*x) + c^6*d^6*(d + e*x)^2 - 2*a*c^5*d^6*e^2 + a^2*c^4*d^4*e^
4) + (2*e^2*(3*c^3*d^4 - 3*a*c^2*d^2*e^2)*(d + e*x)^(1/2))/(c^6*d^6) + (35*e^2*atan((c^(1/2)*d^(1/2)*e^2*(a*e^
2 - c*d^2)^(3/2)*(d + e*x)^(1/2))/(a^2*e^6 + c^2*d^4*e^2 - 2*a*c*d^2*e^4))*(a*e^2 - c*d^2)^(3/2))/(4*c^(9/2)*d
^(9/2))