3.21.18 \(\int \frac {(d+e x)^{15/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [2018]

Optimal. Leaf size=222 \[ \frac {63 e^2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{4 c^5 d^5}+\frac {21 e^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{4 c^4 d^4}+\frac {63 e^2 (d+e x)^{5/2}}{20 c^3 d^3}-\frac {9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{9/2}}{2 c d (a e+c d x)^2}-\frac {63 e^2 \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{11/2} d^{11/2}} \]

[Out]

21/4*e^2*(-a*e^2+c*d^2)*(e*x+d)^(3/2)/c^4/d^4+63/20*e^2*(e*x+d)^(5/2)/c^3/d^3-9/4*e*(e*x+d)^(7/2)/c^2/d^2/(c*d
*x+a*e)-1/2*(e*x+d)^(9/2)/c/d/(c*d*x+a*e)^2-63/4*e^2*(-a*e^2+c*d^2)^(5/2)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2
)/(-a*e^2+c*d^2)^(1/2))/c^(11/2)/d^(11/2)+63/4*e^2*(-a*e^2+c*d^2)^2*(e*x+d)^(1/2)/c^5/d^5

________________________________________________________________________________________

Rubi [A]
time = 0.11, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 43, 52, 65, 214} \begin {gather*} -\frac {63 e^2 \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{11/2} d^{11/2}}+\frac {63 e^2 \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{4 c^5 d^5}+\frac {21 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{4 c^4 d^4}-\frac {9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac {63 e^2 (d+e x)^{5/2}}{20 c^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(15/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(63*e^2*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/(4*c^5*d^5) + (21*e^2*(c*d^2 - a*e^2)*(d + e*x)^(3/2))/(4*c^4*d^4) +
(63*e^2*(d + e*x)^(5/2))/(20*c^3*d^3) - (9*e*(d + e*x)^(7/2))/(4*c^2*d^2*(a*e + c*d*x)) - (d + e*x)^(9/2)/(2*c
*d*(a*e + c*d*x)^2) - (63*e^2*(c*d^2 - a*e^2)^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2
]])/(4*c^(11/2)*d^(11/2))

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*} \int \frac {(d+e x)^{15/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {(d+e x)^{9/2}}{(a e+c d x)^3} \, dx\\ &=-\frac {(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{(a e+c d x)^2} \, dx}{4 c d}\\ &=-\frac {9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac {\left (63 e^2\right ) \int \frac {(d+e x)^{5/2}}{a e+c d x} \, dx}{8 c^2 d^2}\\ &=\frac {63 e^2 (d+e x)^{5/2}}{20 c^3 d^3}-\frac {9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac {\left (63 e^2 \left (c d^2-a e^2\right )\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{8 c^3 d^3}\\ &=\frac {21 e^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{4 c^4 d^4}+\frac {63 e^2 (d+e x)^{5/2}}{20 c^3 d^3}-\frac {9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac {\left (63 e^2 \left (c d^2-a e^2\right )^2\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{8 c^4 d^4}\\ &=\frac {63 e^2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{4 c^5 d^5}+\frac {21 e^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{4 c^4 d^4}+\frac {63 e^2 (d+e x)^{5/2}}{20 c^3 d^3}-\frac {9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac {\left (63 e^2 \left (c d^2-a e^2\right )^3\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{8 c^5 d^5}\\ &=\frac {63 e^2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{4 c^5 d^5}+\frac {21 e^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{4 c^4 d^4}+\frac {63 e^2 (d+e x)^{5/2}}{20 c^3 d^3}-\frac {9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{9/2}}{2 c d (a e+c d x)^2}+\frac {\left (63 e \left (c d^2-a e^2\right )^3\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^5 d^5}\\ &=\frac {63 e^2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{4 c^5 d^5}+\frac {21 e^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{4 c^4 d^4}+\frac {63 e^2 (d+e x)^{5/2}}{20 c^3 d^3}-\frac {9 e (d+e x)^{7/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{9/2}}{2 c d (a e+c d x)^2}-\frac {63 e^2 \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{11/2} d^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.65, size = 250, normalized size = 1.13 \begin {gather*} -\frac {\sqrt {d+e x} \left (-315 a^4 e^8+105 a^3 c d e^6 (7 d-5 e x)-21 a^2 c^2 d^2 e^4 \left (23 d^2-59 d e x+8 e^2 x^2\right )+3 a c^3 d^3 e^2 \left (15 d^3-277 d^2 e x+136 d e^2 x^2+8 e^3 x^3\right )+c^4 d^4 \left (10 d^4+85 d^3 e x-288 d^2 e^2 x^2-56 d e^3 x^3-8 e^4 x^4\right )\right )}{20 c^5 d^5 (a e+c d x)^2}-\frac {63 e^2 \left (-c d^2+a e^2\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{4 c^{11/2} d^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(15/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-1/20*(Sqrt[d + e*x]*(-315*a^4*e^8 + 105*a^3*c*d*e^6*(7*d - 5*e*x) - 21*a^2*c^2*d^2*e^4*(23*d^2 - 59*d*e*x + 8
*e^2*x^2) + 3*a*c^3*d^3*e^2*(15*d^3 - 277*d^2*e*x + 136*d*e^2*x^2 + 8*e^3*x^3) + c^4*d^4*(10*d^4 + 85*d^3*e*x
- 288*d^2*e^2*x^2 - 56*d*e^3*x^3 - 8*e^4*x^4)))/(c^5*d^5*(a*e + c*d*x)^2) - (63*e^2*(-(c*d^2) + a*e^2)^(5/2)*A
rcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(4*c^(11/2)*d^(11/2))

________________________________________________________________________________________

Maple [A]
time = 0.81, size = 342, normalized size = 1.54

method	result	size
derivativedivides	\(2 e^{2} \left (\frac {\frac {c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-a c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}+6 a^{2} e^{4} \sqrt {e x +d}-12 a c \,d^{2} e^{2} \sqrt {e x +d}+6 c^{2} d^{4} \sqrt {e x +d}}{c^{5} d^{5}}-\frac {\frac {\left (-\frac {17}{8} d \,e^{6} c \,a^{3}+\frac {51}{8} d^{3} e^{4} a^{2} c^{2}-\frac {51}{8} d^{5} e^{2} c^{3} a +\frac {17}{8} d^{7} c^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {15}{8} a^{4} e^{8}+\frac {15}{2} a^{3} c \,d^{2} e^{6}-\frac {45}{4} a^{2} c^{2} d^{4} e^{4}+\frac {15}{2} a \,c^{3} d^{6} e^{2}-\frac {15}{8} c^{4} d^{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {63 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{5} d^{5}}\right )\)	\(342\)
default	\(2 e^{2} \left (\frac {\frac {c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-a c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}+6 a^{2} e^{4} \sqrt {e x +d}-12 a c \,d^{2} e^{2} \sqrt {e x +d}+6 c^{2} d^{4} \sqrt {e x +d}}{c^{5} d^{5}}-\frac {\frac {\left (-\frac {17}{8} d \,e^{6} c \,a^{3}+\frac {51}{8} d^{3} e^{4} a^{2} c^{2}-\frac {51}{8} d^{5} e^{2} c^{3} a +\frac {17}{8} d^{7} c^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {15}{8} a^{4} e^{8}+\frac {15}{2} a^{3} c \,d^{2} e^{6}-\frac {45}{4} a^{2} c^{2} d^{4} e^{4}+\frac {15}{2} a \,c^{3} d^{6} e^{2}-\frac {15}{8} c^{4} d^{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {63 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{5} d^{5}}\right )\)	\(342\)
risch	\(\frac {2 e^{2} \left (e^{2} x^{2} c^{2} d^{2}-5 a c d \,e^{3} x +7 c^{2} d^{3} e x +30 a^{2} e^{4}-65 a c \,d^{2} e^{2}+36 c^{2} d^{4}\right ) \sqrt {e x +d}}{5 c^{5} d^{5}}+\frac {17 e^{8} \left (e x +d \right )^{\frac {3}{2}} a^{3}}{4 d^{4} c^{4} \left (c d e x +e^{2} a \right )^{2}}-\frac {51 e^{6} \left (e x +d \right )^{\frac {3}{2}} a^{2}}{4 d^{2} c^{3} \left (c d e x +e^{2} a \right )^{2}}+\frac {51 e^{4} \left (e x +d \right )^{\frac {3}{2}} a}{4 c^{2} \left (c d e x +e^{2} a \right )^{2}}-\frac {17 d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{4 c \left (c d e x +e^{2} a \right )^{2}}+\frac {15 e^{10} \sqrt {e x +d}\, a^{4}}{4 d^{5} c^{5} \left (c d e x +e^{2} a \right )^{2}}-\frac {15 e^{8} \sqrt {e x +d}\, a^{3}}{d^{3} c^{4} \left (c d e x +e^{2} a \right )^{2}}+\frac {45 e^{6} \sqrt {e x +d}\, a^{2}}{2 d \,c^{3} \left (c d e x +e^{2} a \right )^{2}}-\frac {15 d \,e^{4} \sqrt {e x +d}\, a}{c^{2} \left (c d e x +e^{2} a \right )^{2}}+\frac {15 d^{3} e^{2} \sqrt {e x +d}}{4 c \left (c d e x +e^{2} a \right )^{2}}-\frac {63 e^{8} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{3}}{4 d^{5} c^{5} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {189 e^{6} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{2}}{4 d^{3} c^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {189 e^{4} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a}{4 d \,c^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {63 d \,e^{2} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{4 c^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\)	\(599\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(15/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^2*(1/c^5/d^5*(1/5*c^2*d^2*(e*x+d)^(5/2)-a*c*d*e^2*(e*x+d)^(3/2)+c^2*d^3*(e*x+d)^(3/2)+6*a^2*e^4*(e*x+d)^(1
/2)-12*a*c*d^2*e^2*(e*x+d)^(1/2)+6*c^2*d^4*(e*x+d)^(1/2))-1/c^5/d^5*(((-17/8*d*e^6*c*a^3+51/8*d^3*e^4*a^2*c^2-
51/8*d^5*e^2*c^3*a+17/8*d^7*c^4)*(e*x+d)^(3/2)+(-15/8*a^4*e^8+15/2*a^3*c*d^2*e^6-45/4*a^2*c^2*d^4*e^4+15/2*a*c
^3*d^6*e^2-15/8*c^4*d^8)*(e*x+d)^(1/2))/(c*d*(e*x+d)+e^2*a-c*d^2)^2+63/8*(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*
e^2-c^3*d^6)/((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))))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(15/2)/(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(c*d^2-%e^2*a>0)', see `assume?
` for more d

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (187) = 374\).
time = 2.92, size = 841, normalized size = 3.79 \begin {gather*} \left [\frac {315 \, {\left (c^{4} d^{6} x^{2} e^{2} + 2 \, a c^{3} d^{5} x e^{3} - 4 \, a^{2} c^{2} d^{3} x e^{5} + 2 \, a^{3} c d x e^{7} + a^{4} e^{8} + {\left (a^{2} c^{2} d^{2} x^{2} - 2 \, a^{3} c d^{2}\right )} e^{6} - {\left (2 \, a c^{3} d^{4} x^{2} - a^{2} c^{2} d^{4}\right )} e^{4}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d x e + 2 \, c d^{2} - 2 \, \sqrt {x e + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}} - a e^{2}}{c d x + a e}\right ) - 2 \, {\left (85 \, c^{4} d^{7} x e + 10 \, c^{4} d^{8} - 525 \, a^{3} c d x e^{7} - 315 \, a^{4} e^{8} - 21 \, {\left (8 \, a^{2} c^{2} d^{2} x^{2} - 35 \, a^{3} c d^{2}\right )} e^{6} + 3 \, {\left (8 \, a c^{3} d^{3} x^{3} + 413 \, a^{2} c^{2} d^{3} x\right )} e^{5} - {\left (8 \, c^{4} d^{4} x^{4} - 408 \, a c^{3} d^{4} x^{2} + 483 \, a^{2} c^{2} d^{4}\right )} e^{4} - {\left (56 \, c^{4} d^{5} x^{3} + 831 \, a c^{3} d^{5} x\right )} e^{3} - 9 \, {\left (32 \, c^{4} d^{6} x^{2} - 5 \, a c^{3} d^{6}\right )} e^{2}\right )} \sqrt {x e + d}}{40 \, {\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} x e + a^{2} c^{5} d^{5} e^{2}\right )}}, -\frac {315 \, {\left (c^{4} d^{6} x^{2} e^{2} + 2 \, a c^{3} d^{5} x e^{3} - 4 \, a^{2} c^{2} d^{3} x e^{5} + 2 \, a^{3} c d x e^{7} + a^{4} e^{8} + {\left (a^{2} c^{2} d^{2} x^{2} - 2 \, a^{3} c d^{2}\right )} e^{6} - {\left (2 \, a c^{3} d^{4} x^{2} - a^{2} c^{2} d^{4}\right )} e^{4}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {x e + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) + {\left (85 \, c^{4} d^{7} x e + 10 \, c^{4} d^{8} - 525 \, a^{3} c d x e^{7} - 315 \, a^{4} e^{8} - 21 \, {\left (8 \, a^{2} c^{2} d^{2} x^{2} - 35 \, a^{3} c d^{2}\right )} e^{6} + 3 \, {\left (8 \, a c^{3} d^{3} x^{3} + 413 \, a^{2} c^{2} d^{3} x\right )} e^{5} - {\left (8 \, c^{4} d^{4} x^{4} - 408 \, a c^{3} d^{4} x^{2} + 483 \, a^{2} c^{2} d^{4}\right )} e^{4} - {\left (56 \, c^{4} d^{5} x^{3} + 831 \, a c^{3} d^{5} x\right )} e^{3} - 9 \, {\left (32 \, c^{4} d^{6} x^{2} - 5 \, a c^{3} d^{6}\right )} e^{2}\right )} \sqrt {x e + d}}{20 \, {\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} x e + a^{2} c^{5} d^{5} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(15/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

[1/40*(315*(c^4*d^6*x^2*e^2 + 2*a*c^3*d^5*x*e^3 - 4*a^2*c^2*d^3*x*e^5 + 2*a^3*c*d*x*e^7 + a^4*e^8 + (a^2*c^2*d
^2*x^2 - 2*a^3*c*d^2)*e^6 - (2*a*c^3*d^4*x^2 - a^2*c^2*d^4)*e^4)*sqrt((c*d^2 - a*e^2)/(c*d))*log((c*d*x*e + 2*
c*d^2 - 2*sqrt(x*e + d)*c*d*sqrt((c*d^2 - a*e^2)/(c*d)) - a*e^2)/(c*d*x + a*e)) - 2*(85*c^4*d^7*x*e + 10*c^4*d
^8 - 525*a^3*c*d*x*e^7 - 315*a^4*e^8 - 21*(8*a^2*c^2*d^2*x^2 - 35*a^3*c*d^2)*e^6 + 3*(8*a*c^3*d^3*x^3 + 413*a^
2*c^2*d^3*x)*e^5 - (8*c^4*d^4*x^4 - 408*a*c^3*d^4*x^2 + 483*a^2*c^2*d^4)*e^4 - (56*c^4*d^5*x^3 + 831*a*c^3*d^5
*x)*e^3 - 9*(32*c^4*d^6*x^2 - 5*a*c^3*d^6)*e^2)*sqrt(x*e + d))/(c^7*d^7*x^2 + 2*a*c^6*d^6*x*e + a^2*c^5*d^5*e^
2), -1/20*(315*(c^4*d^6*x^2*e^2 + 2*a*c^3*d^5*x*e^3 - 4*a^2*c^2*d^3*x*e^5 + 2*a^3*c*d*x*e^7 + a^4*e^8 + (a^2*c
^2*d^2*x^2 - 2*a^3*c*d^2)*e^6 - (2*a*c^3*d^4*x^2 - a^2*c^2*d^4)*e^4)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(-sqrt
(x*e + d)*c*d*sqrt(-(c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e^2)) + (85*c^4*d^7*x*e + 10*c^4*d^8 - 525*a^3*c*d*x*e^7
- 315*a^4*e^8 - 21*(8*a^2*c^2*d^2*x^2 - 35*a^3*c*d^2)*e^6 + 3*(8*a*c^3*d^3*x^3 + 413*a^2*c^2*d^3*x)*e^5 - (8*
c^4*d^4*x^4 - 408*a*c^3*d^4*x^2 + 483*a^2*c^2*d^4)*e^4 - (56*c^4*d^5*x^3 + 831*a*c^3*d^5*x)*e^3 - 9*(32*c^4*d^
6*x^2 - 5*a*c^3*d^6)*e^2)*sqrt(x*e + d))/(c^7*d^7*x^2 + 2*a*c^6*d^6*x*e + a^2*c^5*d^5*e^2)]

________________________________________________________________________________________

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((e*x+d)**(15/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (187) = 374\).
time = 2.22, size = 413, normalized size = 1.86 \begin {gather*} \frac {63 \, {\left (c^{3} d^{6} e^{2} - 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} - a^{3} e^{8}\right )} \arctan \left (\frac {\sqrt {x e + 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^{5} d^{5}} - \frac {17 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{7} e^{2} - 15 \, \sqrt {x e + d} c^{4} d^{8} e^{2} - 51 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{3} d^{5} e^{4} + 60 \, \sqrt {x e + d} a c^{3} d^{6} e^{4} + 51 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c^{2} d^{3} e^{6} - 90 \, \sqrt {x e + d} a^{2} c^{2} d^{4} e^{6} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} c d e^{8} + 60 \, \sqrt {x e + d} a^{3} c d^{2} e^{8} - 15 \, \sqrt {x e + d} a^{4} e^{10}}{4 \, {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}^{2} c^{5} d^{5}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {5}{2}} c^{12} d^{12} e^{2} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{12} d^{13} e^{2} + 30 \, \sqrt {x e + d} c^{12} d^{14} e^{2} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{11} d^{11} e^{4} - 60 \, \sqrt {x e + d} a c^{11} d^{12} e^{4} + 30 \, \sqrt {x e + d} a^{2} c^{10} d^{10} e^{6}\right )}}{5 \, c^{15} d^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(15/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

63/4*(c^3*d^6*e^2 - 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6 - a^3*e^8)*arctan(sqrt(x*e + d)*c*d/sqrt(-c^2*d^3 + a*c*
d*e^2))/(sqrt(-c^2*d^3 + a*c*d*e^2)*c^5*d^5) - 1/4*(17*(x*e + d)^(3/2)*c^4*d^7*e^2 - 15*sqrt(x*e + d)*c^4*d^8*
e^2 - 51*(x*e + d)^(3/2)*a*c^3*d^5*e^4 + 60*sqrt(x*e + d)*a*c^3*d^6*e^4 + 51*(x*e + d)^(3/2)*a^2*c^2*d^3*e^6 -
90*sqrt(x*e + d)*a^2*c^2*d^4*e^6 - 17*(x*e + d)^(3/2)*a^3*c*d*e^8 + 60*sqrt(x*e + d)*a^3*c*d^2*e^8 - 15*sqrt(
x*e + d)*a^4*e^10)/(((x*e + d)*c*d - c*d^2 + a*e^2)^2*c^5*d^5) + 2/5*((x*e + d)^(5/2)*c^12*d^12*e^2 + 5*(x*e +
d)^(3/2)*c^12*d^13*e^2 + 30*sqrt(x*e + d)*c^12*d^14*e^2 - 5*(x*e + d)^(3/2)*a*c^11*d^11*e^4 - 60*sqrt(x*e + d
)*a*c^11*d^12*e^4 + 30*sqrt(x*e + d)*a^2*c^10*d^10*e^6)/(c^15*d^15)

________________________________________________________________________________________

Mupad [B]
time = 0.73, size = 430, normalized size = 1.94 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {15\,a^4\,e^{10}}{4}-15\,a^3\,c\,d^2\,e^8+\frac {45\,a^2\,c^2\,d^4\,e^6}{2}-15\,a\,c^3\,d^6\,e^4+\frac {15\,c^4\,d^8\,e^2}{4}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (-\frac {17\,a^3\,c\,d\,e^8}{4}+\frac {51\,a^2\,c^2\,d^3\,e^6}{4}-\frac {51\,a\,c^3\,d^5\,e^4}{4}+\frac {17\,c^4\,d^7\,e^2}{4}\right )}{c^7\,d^9-\left (2\,c^7\,d^8-2\,a\,c^6\,d^6\,e^2\right )\,\left (d+e\,x\right )+c^7\,d^7\,{\left (d+e\,x\right )}^2-2\,a\,c^6\,d^7\,e^2+a^2\,c^5\,d^5\,e^4}+\left (\frac {2\,e^2\,{\left (3\,c^3\,d^4-3\,a\,c^2\,d^2\,e^2\right )}^2}{c^9\,d^9}-\frac {6\,e^2\,{\left (a\,e^2-c\,d^2\right )}^2}{c^5\,d^5}\right )\,\sqrt {d+e\,x}+\frac {2\,e^2\,{\left (d+e\,x\right )}^{5/2}}{5\,c^3\,d^3}-\frac {63\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e^2\,{\left (a\,e^2-c\,d^2\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^8-3\,a^2\,c\,d^2\,e^6+3\,a\,c^2\,d^4\,e^4-c^3\,d^6\,e^2}\right )\,{\left (a\,e^2-c\,d^2\right )}^{5/2}}{4\,c^{11/2}\,d^{11/2}}+\frac {2\,e^2\,\left (3\,c^3\,d^4-3\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,c^6\,d^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(15/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

((d + e*x)^(1/2)*((15*a^4*e^10)/4 + (15*c^4*d^8*e^2)/4 - 15*a*c^3*d^6*e^4 - 15*a^3*c*d^2*e^8 + (45*a^2*c^2*d^4
*e^6)/2) - (d + e*x)^(3/2)*((17*c^4*d^7*e^2)/4 - (51*a*c^3*d^5*e^4)/4 + (51*a^2*c^2*d^3*e^6)/4 - (17*a^3*c*d*e
^8)/4))/(c^7*d^9 - (2*c^7*d^8 - 2*a*c^6*d^6*e^2)*(d + e*x) + c^7*d^7*(d + e*x)^2 - 2*a*c^6*d^7*e^2 + a^2*c^5*d
^5*e^4) + ((2*e^2*(3*c^3*d^4 - 3*a*c^2*d^2*e^2)^2)/(c^9*d^9) - (6*e^2*(a*e^2 - c*d^2)^2)/(c^5*d^5))*(d + e*x)^
(1/2) + (2*e^2*(d + e*x)^(5/2))/(5*c^3*d^3) - (63*e^2*atan((c^(1/2)*d^(1/2)*e^2*(a*e^2 - c*d^2)^(5/2)*(d + e*x
)^(1/2))/(a^3*e^8 - c^3*d^6*e^2 + 3*a*c^2*d^4*e^4 - 3*a^2*c*d^2*e^6))*(a*e^2 - c*d^2)^(5/2))/(4*c^(11/2)*d^(11
/2)) + (2*e^2*(3*c^3*d^4 - 3*a*c^2*d^2*e^2)*(d + e*x)^(3/2))/(3*c^6*d^6)

________________________________________________________________________________________