3.21.9 \(\int \frac {(d+e x)^{13/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [2009]

Optimal. Leaf size=210 \[ \frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}-\frac {9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 210, 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 {9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}+\frac {9 e \sqrt {d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}+\frac {3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac {9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*e*(c*d^2 - a*e^2)^3*Sqrt[d + e*x])/(c^5*d^5) + (3*e*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2))/(c^4*d^4) + (9*e*(c*
d^2 - a*e^2)*(d + e*x)^(5/2))/(5*c^3*d^3) + (9*e*(d + e*x)^(7/2))/(7*c^2*d^2) - (d + e*x)^(9/2)/(c*d*(a*e + c*
d*x)) - (9*e*(c*d^2 - a*e^2)^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(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)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {(d+e x)^{9/2}}{(a e+c d x)^2} \, dx\\ &=-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{a e+c d x} \, dx}{2 c d}\\ &=\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )\right ) \int \frac {(d+e x)^{5/2}}{a e+c d x} \, dx}{2 c^2 d^2}\\ &=\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )^2\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{2 c^3 d^3}\\ &=\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )^3\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{2 c^4 d^4}\\ &=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )^4\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 c^5 d^5}\\ &=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 \left (c d^2-a e^2\right )^4\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 )}{c^5 d^5}\\ &=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}-\frac {9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 246, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {d+e x} \left (315 a^4 e^8-210 a^3 c d e^6 (5 d-e x)+42 a^2 c^2 d^2 e^4 \left (29 d^2-17 d e x-e^2 x^2\right )+6 a c^3 d^3 e^2 \left (-88 d^3+142 d^2 e x+23 d e^2 x^2+3 e^3 x^3\right )+c^4 d^4 \left (35 d^4-388 d^3 e x-156 d^2 e^2 x^2-58 d e^3 x^3-10 e^4 x^4\right )\right )}{35 c^5 d^5 (a e+c d x)}+\frac {9 e \left (-c d^2+a e^2\right )^{7/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{11/2} d^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/35*(Sqrt[d + e*x]*(315*a^4*e^8 - 210*a^3*c*d*e^6*(5*d - e*x) + 42*a^2*c^2*d^2*e^4*(29*d^2 - 17*d*e*x - e^2*
x^2) + 6*a*c^3*d^3*e^2*(-88*d^3 + 142*d^2*e*x + 23*d*e^2*x^2 + 3*e^3*x^3) + c^4*d^4*(35*d^4 - 388*d^3*e*x - 15
6*d^2*e^2*x^2 - 58*d*e^3*x^3 - 10*e^4*x^4)))/(c^5*d^5*(a*e + c*d*x)) + (9*e*(-(c*d^2) + a*e^2)^(7/2)*ArcTan[(S
qrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(c^(11/2)*d^(11/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(182)=364\).
time = 0.91, size = 375, normalized size = 1.79

method	result	size
derivativedivides	\(2 e \left (-\frac {-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 c^{3} d^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}-a^{2} c d \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+2 a \,c^{2} d^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}-c^{3} d^{5} \left (e x +d \right )^{\frac {3}{2}}+4 a^{3} e^{6} \sqrt {e x +d}-12 a^{2} e^{4} d^{2} c \sqrt {e x +d}+12 a \,e^{2} d^{4} c^{2} \sqrt {e x +d}-4 d^{6} c^{3} \sqrt {e x +d}}{c^{5} d^{5}}+\frac {\frac {\left (-\frac {1}{2} a^{4} e^{8}+2 a^{3} c \,d^{2} e^{6}-3 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}-\frac {1}{2} c^{4} d^{8}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {9 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{d^{5} c^{5}}\right )\)	\(375\)
default	\(2 e \left (-\frac {-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 c^{3} d^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}-a^{2} c d \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+2 a \,c^{2} d^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}-c^{3} d^{5} \left (e x +d \right )^{\frac {3}{2}}+4 a^{3} e^{6} \sqrt {e x +d}-12 a^{2} e^{4} d^{2} c \sqrt {e x +d}+12 a \,e^{2} d^{4} c^{2} \sqrt {e x +d}-4 d^{6} c^{3} \sqrt {e x +d}}{c^{5} d^{5}}+\frac {\frac {\left (-\frac {1}{2} a^{4} e^{8}+2 a^{3} c \,d^{2} e^{6}-3 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}-\frac {1}{2} c^{4} d^{8}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {9 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{d^{5} c^{5}}\right )\)	\(375\)
risch	\(-\frac {2 e \left (-5 c^{3} d^{3} e^{3} x^{3}+14 a \,c^{2} d^{2} e^{4} x^{2}-29 c^{3} d^{4} e^{2} x^{2}-35 a^{2} c d \,e^{5} x +98 a \,c^{2} d^{3} e^{3} x -78 c^{3} d^{5} e x +140 e^{6} a^{3}-455 e^{4} d^{2} a^{2} c +504 d^{4} e^{2} c^{2} a -194 d^{6} c^{3}\right ) \sqrt {e x +d}}{35 c^{5} d^{5}}-\frac {e^{9} \sqrt {e x +d}\, a^{4}}{d^{5} c^{5} \left (c d e x +e^{2} a \right )}+\frac {4 e^{7} \sqrt {e x +d}\, a^{3}}{d^{3} c^{4} \left (c d e x +e^{2} a \right )}-\frac {6 e^{5} \sqrt {e x +d}\, a^{2}}{d \,c^{3} \left (c d e x +e^{2} a \right )}+\frac {4 d \,e^{3} \sqrt {e x +d}\, a}{c^{2} \left (c d e x +e^{2} a \right )}-\frac {d^{3} e \sqrt {e x +d}}{c \left (c d e x +e^{2} a \right )}+\frac {9 e^{9} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{4}}{d^{5} c^{5} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {36 e^{7} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{3}}{d^{3} c^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {54 e^{5} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{2}}{d \,c^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {36 d \,e^{3} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a}{c^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {9 d^{3} e \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\)	\(583\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e*(-1/c^5/d^5*(-1/7*c^3*d^3*(e*x+d)^(7/2)+2/5*a*c^2*d^2*e^2*(e*x+d)^(5/2)-2/5*c^3*d^4*(e*x+d)^(5/2)-a^2*c*d*
e^4*(e*x+d)^(3/2)+2*a*c^2*d^3*e^2*(e*x+d)^(3/2)-c^3*d^5*(e*x+d)^(3/2)+4*a^3*e^6*(e*x+d)^(1/2)-12*a^2*e^4*d^2*c
*(e*x+d)^(1/2)+12*a*e^2*d^4*c^2*(e*x+d)^(1/2)-4*d^6*c^3*(e*x+d)^(1/2))+1/d^5/c^5*((-1/2*a^4*e^8+2*a^3*c*d^2*e^
6-3*a^2*c^2*d^4*e^4+2*a*c^3*d^6*e^2-1/2*c^4*d^8)*(e*x+d)^(1/2)/(c*d*(e*x+d)+e^2*a-c*d^2)+9/2*(a^4*e^8-4*a^3*c*
d^2*e^6+6*a^2*c^2*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^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))))

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

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Fricas [A]
time = 4.36, size = 766, normalized size = 3.65 \begin {gather*} \left [\frac {315 \, {\left (c^{4} d^{7} x e - 3 \, a c^{3} d^{5} x e^{3} + a c^{3} d^{6} e^{2} + 3 \, a^{2} c^{2} d^{3} x e^{5} - 3 \, a^{2} c^{2} d^{4} e^{4} - a^{3} c d x e^{7} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\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 (388 \, c^{4} d^{7} x e - 35 \, c^{4} d^{8} - 210 \, a^{3} c d x e^{7} - 315 \, a^{4} e^{8} + 42 \, {\left (a^{2} c^{2} d^{2} x^{2} + 25 \, a^{3} c d^{2}\right )} e^{6} - 6 \, {\left (3 \, a c^{3} d^{3} x^{3} - 119 \, a^{2} c^{2} d^{3} x\right )} e^{5} + 2 \, {\left (5 \, c^{4} d^{4} x^{4} - 69 \, a c^{3} d^{4} x^{2} - 609 \, a^{2} c^{2} d^{4}\right )} e^{4} + 2 \, {\left (29 \, c^{4} d^{5} x^{3} - 426 \, a c^{3} d^{5} x\right )} e^{3} + 12 \, {\left (13 \, c^{4} d^{6} x^{2} + 44 \, a c^{3} d^{6}\right )} e^{2}\right )} \sqrt {x e + d}}{70 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}, -\frac {315 \, {\left (c^{4} d^{7} x e - 3 \, a c^{3} d^{5} x e^{3} + a c^{3} d^{6} e^{2} + 3 \, a^{2} c^{2} d^{3} x e^{5} - 3 \, a^{2} c^{2} d^{4} e^{4} - a^{3} c d x e^{7} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\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 (388 \, c^{4} d^{7} x e - 35 \, c^{4} d^{8} - 210 \, a^{3} c d x e^{7} - 315 \, a^{4} e^{8} + 42 \, {\left (a^{2} c^{2} d^{2} x^{2} + 25 \, a^{3} c d^{2}\right )} e^{6} - 6 \, {\left (3 \, a c^{3} d^{3} x^{3} - 119 \, a^{2} c^{2} d^{3} x\right )} e^{5} + 2 \, {\left (5 \, c^{4} d^{4} x^{4} - 69 \, a c^{3} d^{4} x^{2} - 609 \, a^{2} c^{2} d^{4}\right )} e^{4} + 2 \, {\left (29 \, c^{4} d^{5} x^{3} - 426 \, a c^{3} d^{5} x\right )} e^{3} + 12 \, {\left (13 \, c^{4} d^{6} x^{2} + 44 \, a c^{3} d^{6}\right )} e^{2}\right )} \sqrt {x e + d}}{35 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/70*(315*(c^4*d^7*x*e - 3*a*c^3*d^5*x*e^3 + a*c^3*d^6*e^2 + 3*a^2*c^2*d^3*x*e^5 - 3*a^2*c^2*d^4*e^4 - a^3*c*
d*x*e^7 + 3*a^3*c*d^2*e^6 - a^4*e^8)*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*(388*c^4*d^7*x*e - 35*c^4*d^8 - 210*a^3*c*d*x*e^7 - 31
5*a^4*e^8 + 42*(a^2*c^2*d^2*x^2 + 25*a^3*c*d^2)*e^6 - 6*(3*a*c^3*d^3*x^3 - 119*a^2*c^2*d^3*x)*e^5 + 2*(5*c^4*d
^4*x^4 - 69*a*c^3*d^4*x^2 - 609*a^2*c^2*d^4)*e^4 + 2*(29*c^4*d^5*x^3 - 426*a*c^3*d^5*x)*e^3 + 12*(13*c^4*d^6*x
^2 + 44*a*c^3*d^6)*e^2)*sqrt(x*e + d))/(c^6*d^6*x + a*c^5*d^5*e), -1/35*(315*(c^4*d^7*x*e - 3*a*c^3*d^5*x*e^3
+ a*c^3*d^6*e^2 + 3*a^2*c^2*d^3*x*e^5 - 3*a^2*c^2*d^4*e^4 - a^3*c*d*x*e^7 + 3*a^3*c*d^2*e^6 - a^4*e^8)*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)) - (388*c^4*d^7*x
*e - 35*c^4*d^8 - 210*a^3*c*d*x*e^7 - 315*a^4*e^8 + 42*(a^2*c^2*d^2*x^2 + 25*a^3*c*d^2)*e^6 - 6*(3*a*c^3*d^3*x
^3 - 119*a^2*c^2*d^3*x)*e^5 + 2*(5*c^4*d^4*x^4 - 69*a*c^3*d^4*x^2 - 609*a^2*c^2*d^4)*e^4 + 2*(29*c^4*d^5*x^3 -
426*a*c^3*d^5*x)*e^3 + 12*(13*c^4*d^6*x^2 + 44*a*c^3*d^6)*e^2)*sqrt(x*e + d))/(c^6*d^6*x + a*c^5*d^5*e)]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (189) = 378\).
time = 1.84, size = 428, normalized size = 2.04 \begin {gather*} \frac {9 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{5} d^{5}} - \frac {\sqrt {x e + d} c^{4} d^{8} e - 4 \, \sqrt {x e + d} a c^{3} d^{6} e^{3} + 6 \, \sqrt {x e + d} a^{2} c^{2} d^{4} e^{5} - 4 \, \sqrt {x e + d} a^{3} c d^{2} e^{7} + \sqrt {x e + d} a^{4} e^{9}}{{\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )} c^{5} d^{5}} + \frac {2 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{12} d^{12} e + 14 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{12} d^{13} e + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{12} d^{14} e + 140 \, \sqrt {x e + d} c^{12} d^{15} e - 14 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{11} d^{11} e^{3} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{11} d^{12} e^{3} - 420 \, \sqrt {x e + d} a c^{11} d^{13} e^{3} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c^{10} d^{10} e^{5} + 420 \, \sqrt {x e + d} a^{2} c^{10} d^{11} e^{5} - 140 \, \sqrt {x e + d} a^{3} c^{9} d^{9} e^{7}\right )}}{35 \, c^{14} d^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*(c^4*d^8*e - 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 - 4*a^3*c*d^2*e^7 + a^4*e^9)*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) - (sqrt(x*e + d)*c^4*d^8*e - 4*sqrt(x*e + d)*a*c^3
*d^6*e^3 + 6*sqrt(x*e + d)*a^2*c^2*d^4*e^5 - 4*sqrt(x*e + d)*a^3*c*d^2*e^7 + sqrt(x*e + d)*a^4*e^9)/(((x*e + d
)*c*d - c*d^2 + a*e^2)*c^5*d^5) + 2/35*(5*(x*e + d)^(7/2)*c^12*d^12*e + 14*(x*e + d)^(5/2)*c^12*d^13*e + 35*(x
*e + d)^(3/2)*c^12*d^14*e + 140*sqrt(x*e + d)*c^12*d^15*e - 14*(x*e + d)^(5/2)*a*c^11*d^11*e^3 - 70*(x*e + d)^
(3/2)*a*c^11*d^12*e^3 - 420*sqrt(x*e + d)*a*c^11*d^13*e^3 + 35*(x*e + d)^(3/2)*a^2*c^10*d^10*e^5 + 420*sqrt(x*
e + d)*a^2*c^10*d^11*e^5 - 140*sqrt(x*e + d)*a^3*c^9*d^9*e^7)/(c^14*d^14)

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Mupad [B]
time = 0.70, size = 443, normalized size = 2.11 \begin {gather*} \frac {2\,e\,{\left (d+e\,x\right )}^{7/2}}{7\,c^2\,d^2}-\left (\frac {\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{c^6\,d^6}\right )}{c^2\,d^2}+\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}{c^6\,d^6}\right )\,\sqrt {d+e\,x}-\frac {\sqrt {d+e\,x}\,\left (a^4\,e^9-4\,a^3\,c\,d^2\,e^7+6\,a^2\,c^2\,d^4\,e^5-4\,a\,c^3\,d^6\,e^3+c^4\,d^8\,e\right )}{c^6\,d^6\,\left (d+e\,x\right )-c^6\,d^7+a\,c^5\,d^5\,e^2}-\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{3\,c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{3\,c^6\,d^6}\right )\,{\left (d+e\,x\right )}^{3/2}+\frac {2\,e\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,c^4\,d^4}+\frac {9\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,{\left (a\,e^2-c\,d^2\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^9-4\,a^3\,c\,d^2\,e^7+6\,a^2\,c^2\,d^4\,e^5-4\,a\,c^3\,d^6\,e^3+c^4\,d^8\,e}\right )\,{\left (a\,e^2-c\,d^2\right )}^{7/2}}{c^{11/2}\,d^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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