3.17.66 \(\int \frac {(d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1666]

Optimal. Leaf size=197 \[ \frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}-\frac {693 e^5 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2}} \]

[Out]

-231/128*e^4*(e*x+d)^(3/2)/b^5/(b*x+a)-231/320*e^3*(e*x+d)^(5/2)/b^4/(b*x+a)^2-33/80*e^2*(e*x+d)^(7/2)/b^3/(b*
x+a)^3-11/40*e*(e*x+d)^(9/2)/b^2/(b*x+a)^4-1/5*(e*x+d)^(11/2)/b/(b*x+a)^5-693/128*e^5*arctanh(b^(1/2)*(e*x+d)^
(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(13/2)+693/128*e^5*(e*x+d)^(1/2)/b^6

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Rubi [A]
time = 0.07, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 52, 65, 214} \begin {gather*} -\frac {693 e^5 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2}}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {693 e^5 \sqrt {d+e x}}{128 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(693*e^5*Sqrt[d + e*x])/(128*b^6) - (231*e^4*(d + e*x)^(3/2))/(128*b^5*(a + b*x)) - (231*e^3*(d + e*x)^(5/2))/
(320*b^4*(a + b*x)^2) - (33*e^2*(d + e*x)^(7/2))/(80*b^3*(a + b*x)^3) - (11*e*(d + e*x)^(9/2))/(40*b^2*(a + b*
x)^4) - (d + e*x)^(11/2)/(5*b*(a + b*x)^5) - (693*e^5*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(128*b^(13/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^{11/2}}{(a+b x)^6} \, dx\\ &=-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {(11 e) \int \frac {(d+e x)^{9/2}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (99 e^2\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^4} \, dx}{80 b^2}\\ &=-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (231 e^3\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{160 b^3}\\ &=-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (231 e^4\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{128 b^4}\\ &=-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (693 e^5\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^5}\\ &=\frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (693 e^5 (b d-a e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^6}\\ &=\frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (693 e^4 (b d-a e)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^6}\\ &=\frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}-\frac {693 e^5 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 1.67, size = 273, normalized size = 1.39 \begin {gather*} -\frac {\sqrt {d+e x} \left (-3465 a^5 e^5+1155 a^4 b e^4 (d-14 e x)+462 a^3 b^2 e^3 \left (d^2+12 d e x-64 e^2 x^2\right )+66 a^2 b^3 e^2 \left (4 d^3+33 d^2 e x+159 d e^2 x^2-395 e^3 x^3\right )+11 a b^4 e \left (16 d^4+112 d^3 e x+366 d^2 e^2 x^2+880 d e^3 x^3-965 e^4 x^4\right )+b^5 \left (128 d^5+816 d^4 e x+2248 d^3 e^2 x^2+3590 d^2 e^3 x^3+4215 d e^4 x^4-1280 e^5 x^5\right )\right )}{640 b^6 (a+b x)^5}-\frac {693 e^5 \sqrt {-b d+a e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/640*(Sqrt[d + e*x]*(-3465*a^5*e^5 + 1155*a^4*b*e^4*(d - 14*e*x) + 462*a^3*b^2*e^3*(d^2 + 12*d*e*x - 64*e^2*
x^2) + 66*a^2*b^3*e^2*(4*d^3 + 33*d^2*e*x + 159*d*e^2*x^2 - 395*e^3*x^3) + 11*a*b^4*e*(16*d^4 + 112*d^3*e*x +
366*d^2*e^2*x^2 + 880*d*e^3*x^3 - 965*e^4*x^4) + b^5*(128*d^5 + 816*d^4*e*x + 2248*d^3*e^2*x^2 + 3590*d^2*e^3*
x^3 + 4215*d*e^4*x^4 - 1280*e^5*x^5)))/(b^6*(a + b*x)^5) - (693*e^5*Sqrt[-(b*d) + a*e]*ArcTan[(Sqrt[b]*Sqrt[d
+ e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(13/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(326\) vs. \(2(161)=322\).
time = 0.75, size = 327, normalized size = 1.66

method	result	size
derivativedivides	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{b^{6}}-\frac {\frac {\left (-\frac {843}{256} a \,b^{4} e +\frac {843}{256} b^{5} d \right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {1327 b^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {131}{10} a^{3} b^{2} e^{3}+\frac {393}{10} a^{2} b^{3} d \,e^{2}-\frac {393}{10} a \,b^{4} d^{2} e +\frac {131}{10} b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {977}{128} a^{4} b \,e^{4}+\frac {977}{32} a^{3} b^{2} d \,e^{3}-\frac {2931}{64} a^{2} b^{3} d^{2} e^{2}+\frac {977}{32} a \,b^{4} d^{3} e -\frac {977}{128} b^{5} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {437}{256} a^{5} e^{5}+\frac {2185}{256} a^{4} b d \,e^{4}-\frac {2185}{128} a^{3} b^{2} d^{2} e^{3}+\frac {2185}{128} a^{2} b^{3} d^{3} e^{2}-\frac {2185}{256} a \,b^{4} d^{4} e +\frac {437}{256} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {693 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}}{b^{6}}\right )\)	\(327\)
default	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{b^{6}}-\frac {\frac {\left (-\frac {843}{256} a \,b^{4} e +\frac {843}{256} b^{5} d \right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {1327 b^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {131}{10} a^{3} b^{2} e^{3}+\frac {393}{10} a^{2} b^{3} d \,e^{2}-\frac {393}{10} a \,b^{4} d^{2} e +\frac {131}{10} b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {977}{128} a^{4} b \,e^{4}+\frac {977}{32} a^{3} b^{2} d \,e^{3}-\frac {2931}{64} a^{2} b^{3} d^{2} e^{2}+\frac {977}{32} a \,b^{4} d^{3} e -\frac {977}{128} b^{5} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {437}{256} a^{5} e^{5}+\frac {2185}{256} a^{4} b d \,e^{4}-\frac {2185}{128} a^{3} b^{2} d^{2} e^{3}+\frac {2185}{128} a^{2} b^{3} d^{3} e^{2}-\frac {2185}{256} a \,b^{4} d^{4} e +\frac {437}{256} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {693 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}}{b^{6}}\right )\)	\(327\)
risch	\(\frac {2 e^{5} \sqrt {e x +d}}{b^{6}}-\frac {977 e^{6} \left (e x +d \right )^{\frac {3}{2}} d^{3} a}{16 b^{2} \left (b e x +a e \right )^{5}}-\frac {2185 e^{9} \sqrt {e x +d}\, a^{4} d}{128 b^{5} \left (b e x +a e \right )^{5}}+\frac {2185 e^{8} \sqrt {e x +d}\, a^{3} d^{2}}{64 b^{4} \left (b e x +a e \right )^{5}}-\frac {2185 e^{7} \sqrt {e x +d}\, a^{2} d^{3}}{64 b^{3} \left (b e x +a e \right )^{5}}+\frac {2185 e^{6} \sqrt {e x +d}\, a \,d^{4}}{128 b^{2} \left (b e x +a e \right )^{5}}-\frac {843 e^{5} \left (e x +d \right )^{\frac {9}{2}} d}{128 b \left (b e x +a e \right )^{5}}+\frac {1327 e^{5} \left (e x +d \right )^{\frac {7}{2}} d^{2}}{64 b \left (b e x +a e \right )^{5}}-\frac {131 e^{5} \left (e x +d \right )^{\frac {5}{2}} d^{3}}{5 b \left (b e x +a e \right )^{5}}+\frac {977 e^{5} \left (e x +d \right )^{\frac {3}{2}} d^{4}}{64 b \left (b e x +a e \right )^{5}}+\frac {437 e^{10} \sqrt {e x +d}\, a^{5}}{128 b^{6} \left (b e x +a e \right )^{5}}-\frac {437 e^{5} \sqrt {e x +d}\, d^{5}}{128 b \left (b e x +a e \right )^{5}}-\frac {693 e^{6} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a}{128 b^{6} \sqrt {b \left (a e -b d \right )}}+\frac {693 e^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d}{128 b^{5} \sqrt {b \left (a e -b d \right )}}-\frac {1327 e^{6} \left (e x +d \right )^{\frac {7}{2}} a d}{32 b^{2} \left (b e x +a e \right )^{5}}-\frac {393 e^{7} \left (e x +d \right )^{\frac {5}{2}} a^{2} d}{5 b^{3} \left (b e x +a e \right )^{5}}+\frac {393 e^{6} \left (e x +d \right )^{\frac {5}{2}} a \,d^{2}}{5 b^{2} \left (b e x +a e \right )^{5}}-\frac {977 e^{8} \left (e x +d \right )^{\frac {3}{2}} a^{3} d}{16 b^{4} \left (b e x +a e \right )^{5}}+\frac {2931 e^{7} \left (e x +d \right )^{\frac {3}{2}} a^{2} d^{2}}{32 b^{3} \left (b e x +a e \right )^{5}}+\frac {843 e^{6} \left (e x +d \right )^{\frac {9}{2}} a}{128 b^{2} \left (b e x +a e \right )^{5}}+\frac {1327 e^{7} \left (e x +d \right )^{\frac {7}{2}} a^{2}}{64 b^{3} \left (b e x +a e \right )^{5}}+\frac {131 e^{8} \left (e x +d \right )^{\frac {5}{2}} a^{3}}{5 b^{4} \left (b e x +a e \right )^{5}}+\frac {977 e^{9} \left (e x +d \right )^{\frac {3}{2}} a^{4}}{64 b^{5} \left (b e x +a e \right )^{5}}\)	\(673\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*(1/b^6*(e*x+d)^(1/2)-1/b^6*(((-843/256*a*b^4*e+843/256*b^5*d)*(e*x+d)^(9/2)-1327/128*b^3*(a^2*e^2-2*a*b*
d*e+b^2*d^2)*(e*x+d)^(7/2)+(-131/10*a^3*b^2*e^3+393/10*a^2*b^3*d*e^2-393/10*a*b^4*d^2*e+131/10*b^5*d^3)*(e*x+d
)^(5/2)+(-977/128*a^4*b*e^4+977/32*a^3*b^2*d*e^3-2931/64*a^2*b^3*d^2*e^2+977/32*a*b^4*d^3*e-977/128*b^5*d^4)*(
e*x+d)^(3/2)+(-437/256*a^5*e^5+2185/256*a^4*b*d*e^4-2185/128*a^3*b^2*d^2*e^3+2185/128*a^2*b^3*d^3*e^2-2185/256
*a*b^4*d^4*e+437/256*b^5*d^5)*(e*x+d)^(1/2))/((e*x+d)*b+a*e-b*d)^5+693/256*(a*e-b*d)/(b*(a*e-b*d))^(1/2)*arcta
n(b*(e*x+d)^(1/2)/(b*(a*e-b*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)^(11/2)/(b^2*x^2+2*a*b*x+a^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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (166) = 332\).
time = 2.34, size = 822, normalized size = 4.17 \begin {gather*} \left [\frac {3465 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \sqrt {\frac {b d - a e}{b}} e^{5} \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (128 \, b^{5} d^{5} - {\left (1280 \, b^{5} x^{5} + 10615 \, a b^{4} x^{4} + 26070 \, a^{2} b^{3} x^{3} + 29568 \, a^{3} b^{2} x^{2} + 16170 \, a^{4} b x + 3465 \, a^{5}\right )} e^{5} + {\left (4215 \, b^{5} d x^{4} + 9680 \, a b^{4} d x^{3} + 10494 \, a^{2} b^{3} d x^{2} + 5544 \, a^{3} b^{2} d x + 1155 \, a^{4} b d\right )} e^{4} + 2 \, {\left (1795 \, b^{5} d^{2} x^{3} + 2013 \, a b^{4} d^{2} x^{2} + 1089 \, a^{2} b^{3} d^{2} x + 231 \, a^{3} b^{2} d^{2}\right )} e^{3} + 8 \, {\left (281 \, b^{5} d^{3} x^{2} + 154 \, a b^{4} d^{3} x + 33 \, a^{2} b^{3} d^{3}\right )} e^{2} + 16 \, {\left (51 \, b^{5} d^{4} x + 11 \, a b^{4} d^{4}\right )} e\right )} \sqrt {x e + d}}{1280 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}, -\frac {3465 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) e^{5} + {\left (128 \, b^{5} d^{5} - {\left (1280 \, b^{5} x^{5} + 10615 \, a b^{4} x^{4} + 26070 \, a^{2} b^{3} x^{3} + 29568 \, a^{3} b^{2} x^{2} + 16170 \, a^{4} b x + 3465 \, a^{5}\right )} e^{5} + {\left (4215 \, b^{5} d x^{4} + 9680 \, a b^{4} d x^{3} + 10494 \, a^{2} b^{3} d x^{2} + 5544 \, a^{3} b^{2} d x + 1155 \, a^{4} b d\right )} e^{4} + 2 \, {\left (1795 \, b^{5} d^{2} x^{3} + 2013 \, a b^{4} d^{2} x^{2} + 1089 \, a^{2} b^{3} d^{2} x + 231 \, a^{3} b^{2} d^{2}\right )} e^{3} + 8 \, {\left (281 \, b^{5} d^{3} x^{2} + 154 \, a b^{4} d^{3} x + 33 \, a^{2} b^{3} d^{3}\right )} e^{2} + 16 \, {\left (51 \, b^{5} d^{4} x + 11 \, a b^{4} d^{4}\right )} e\right )} \sqrt {x e + d}}{640 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(3465*(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5)*sqrt((b*d - a*e)/b)*
e^5*log((2*b*d - 2*sqrt(x*e + d)*b*sqrt((b*d - a*e)/b) + (b*x - a)*e)/(b*x + a)) - 2*(128*b^5*d^5 - (1280*b^5*
x^5 + 10615*a*b^4*x^4 + 26070*a^2*b^3*x^3 + 29568*a^3*b^2*x^2 + 16170*a^4*b*x + 3465*a^5)*e^5 + (4215*b^5*d*x^
4 + 9680*a*b^4*d*x^3 + 10494*a^2*b^3*d*x^2 + 5544*a^3*b^2*d*x + 1155*a^4*b*d)*e^4 + 2*(1795*b^5*d^2*x^3 + 2013
*a*b^4*d^2*x^2 + 1089*a^2*b^3*d^2*x + 231*a^3*b^2*d^2)*e^3 + 8*(281*b^5*d^3*x^2 + 154*a*b^4*d^3*x + 33*a^2*b^3
*d^3)*e^2 + 16*(51*b^5*d^4*x + 11*a*b^4*d^4)*e)*sqrt(x*e + d))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*
a^3*b^8*x^2 + 5*a^4*b^7*x + a^5*b^6), -1/640*(3465*(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 +
5*a^4*b*x + a^5)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e))*e^5 + (128*b^5
*d^5 - (1280*b^5*x^5 + 10615*a*b^4*x^4 + 26070*a^2*b^3*x^3 + 29568*a^3*b^2*x^2 + 16170*a^4*b*x + 3465*a^5)*e^5
+ (4215*b^5*d*x^4 + 9680*a*b^4*d*x^3 + 10494*a^2*b^3*d*x^2 + 5544*a^3*b^2*d*x + 1155*a^4*b*d)*e^4 + 2*(1795*b
^5*d^2*x^3 + 2013*a*b^4*d^2*x^2 + 1089*a^2*b^3*d^2*x + 231*a^3*b^2*d^2)*e^3 + 8*(281*b^5*d^3*x^2 + 154*a*b^4*d
^3*x + 33*a^2*b^3*d^3)*e^2 + 16*(51*b^5*d^4*x + 11*a*b^4*d^4)*e)*sqrt(x*e + d))/(b^11*x^5 + 5*a*b^10*x^4 + 10*
a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^7*x + a^5*b^6)]

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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)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (166) = 332\).
time = 0.76, size = 459, normalized size = 2.33 \begin {gather*} \frac {693 \, {\left (b d e^{5} - a e^{6}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{6}} + \frac {2 \, \sqrt {x e + d} e^{5}}{b^{6}} - \frac {4215 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{5} d e^{5} - 13270 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} d^{2} e^{5} + 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{5} - 9770 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{5} + 2185 \, \sqrt {x e + d} b^{5} d^{5} e^{5} - 4215 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{4} e^{6} + 26540 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{4} d e^{6} - 50304 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{6} + 39080 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{6} - 10925 \, \sqrt {x e + d} a b^{4} d^{4} e^{6} - 13270 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{3} e^{7} + 50304 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{7} - 58620 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{7} + 21850 \, \sqrt {x e + d} a^{2} b^{3} d^{3} e^{7} - 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{8} + 39080 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{8} - 21850 \, \sqrt {x e + d} a^{3} b^{2} d^{2} e^{8} - 9770 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b e^{9} + 10925 \, \sqrt {x e + d} a^{4} b d e^{9} - 2185 \, \sqrt {x e + d} a^{5} e^{10}}{640 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

693/128*(b*d*e^5 - a*e^6)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6) + 2*sqrt(x*e
+ d)*e^5/b^6 - 1/640*(4215*(x*e + d)^(9/2)*b^5*d*e^5 - 13270*(x*e + d)^(7/2)*b^5*d^2*e^5 + 16768*(x*e + d)^(5
/2)*b^5*d^3*e^5 - 9770*(x*e + d)^(3/2)*b^5*d^4*e^5 + 2185*sqrt(x*e + d)*b^5*d^5*e^5 - 4215*(x*e + d)^(9/2)*a*b
^4*e^6 + 26540*(x*e + d)^(7/2)*a*b^4*d*e^6 - 50304*(x*e + d)^(5/2)*a*b^4*d^2*e^6 + 39080*(x*e + d)^(3/2)*a*b^4
*d^3*e^6 - 10925*sqrt(x*e + d)*a*b^4*d^4*e^6 - 13270*(x*e + d)^(7/2)*a^2*b^3*e^7 + 50304*(x*e + d)^(5/2)*a^2*b
^3*d*e^7 - 58620*(x*e + d)^(3/2)*a^2*b^3*d^2*e^7 + 21850*sqrt(x*e + d)*a^2*b^3*d^3*e^7 - 16768*(x*e + d)^(5/2)
*a^3*b^2*e^8 + 39080*(x*e + d)^(3/2)*a^3*b^2*d*e^8 - 21850*sqrt(x*e + d)*a^3*b^2*d^2*e^8 - 9770*(x*e + d)^(3/2
)*a^4*b*e^9 + 10925*sqrt(x*e + d)*a^4*b*d*e^9 - 2185*sqrt(x*e + d)*a^5*e^10)/(((x*e + d)*b - b*d + a*e)^5*b^6)

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Mupad [B]
time = 0.32, size = 598, normalized size = 3.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (\frac {1327\,a^2\,b^3\,e^7}{64}-\frac {1327\,a\,b^4\,d\,e^6}{32}+\frac {1327\,b^5\,d^2\,e^5}{64}\right )+\sqrt {d+e\,x}\,\left (\frac {437\,a^5\,e^{10}}{128}-\frac {2185\,a^4\,b\,d\,e^9}{128}+\frac {2185\,a^3\,b^2\,d^2\,e^8}{64}-\frac {2185\,a^2\,b^3\,d^3\,e^7}{64}+\frac {2185\,a\,b^4\,d^4\,e^6}{128}-\frac {437\,b^5\,d^5\,e^5}{128}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {131\,a^3\,b^2\,e^8}{5}-\frac {393\,a^2\,b^3\,d\,e^7}{5}+\frac {393\,a\,b^4\,d^2\,e^6}{5}-\frac {131\,b^5\,d^3\,e^5}{5}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {977\,a^4\,b\,e^9}{64}-\frac {977\,a^3\,b^2\,d\,e^8}{16}+\frac {2931\,a^2\,b^3\,d^2\,e^7}{32}-\frac {977\,a\,b^4\,d^3\,e^6}{16}+\frac {977\,b^5\,d^4\,e^5}{64}\right )+\left (\frac {843\,a\,b^4\,e^6}{128}-\frac {843\,b^5\,d\,e^5}{128}\right )\,{\left (d+e\,x\right )}^{9/2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b^7\,e^4-20\,a^3\,b^8\,d\,e^3+30\,a^2\,b^9\,d^2\,e^2-20\,a\,b^{10}\,d^3\,e+5\,b^{11}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^8\,e^3+30\,a^2\,b^9\,d\,e^2-30\,a\,b^{10}\,d^2\,e+10\,b^{11}\,d^3\right )+b^{11}\,{\left (d+e\,x\right )}^5-\left (5\,b^{11}\,d-5\,a\,b^{10}\,e\right )\,{\left (d+e\,x\right )}^4-b^{11}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^9\,e^2-20\,a\,b^{10}\,d\,e+10\,b^{11}\,d^2\right )+a^5\,b^6\,e^5-5\,a^4\,b^7\,d\,e^4-10\,a^2\,b^9\,d^3\,e^2+10\,a^3\,b^8\,d^2\,e^3+5\,a\,b^{10}\,d^4\,e}+\frac {2\,e^5\,\sqrt {d+e\,x}}{b^6}-\frac {693\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^5\,\sqrt {a\,e-b\,d}\,\sqrt {d+e\,x}}{a\,e^6-b\,d\,e^5}\right )\,\sqrt {a\,e-b\,d}}{128\,b^{13/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(11/2)/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((d + e*x)^(7/2)*((1327*a^2*b^3*e^7)/64 + (1327*b^5*d^2*e^5)/64 - (1327*a*b^4*d*e^6)/32) + (d + e*x)^(1/2)*((4
37*a^5*e^10)/128 - (437*b^5*d^5*e^5)/128 + (2185*a*b^4*d^4*e^6)/128 - (2185*a^2*b^3*d^3*e^7)/64 + (2185*a^3*b^
2*d^2*e^8)/64 - (2185*a^4*b*d*e^9)/128) + (d + e*x)^(5/2)*((131*a^3*b^2*e^8)/5 - (131*b^5*d^3*e^5)/5 + (393*a*
b^4*d^2*e^6)/5 - (393*a^2*b^3*d*e^7)/5) + (d + e*x)^(3/2)*((977*a^4*b*e^9)/64 + (977*b^5*d^4*e^5)/64 - (977*a*
b^4*d^3*e^6)/16 - (977*a^3*b^2*d*e^8)/16 + (2931*a^2*b^3*d^2*e^7)/32) + ((843*a*b^4*e^6)/128 - (843*b^5*d*e^5)
/128)*(d + e*x)^(9/2))/((d + e*x)*(5*b^11*d^4 + 5*a^4*b^7*e^4 - 20*a^3*b^8*d*e^3 + 30*a^2*b^9*d^2*e^2 - 20*a*b
^10*d^3*e) - (d + e*x)^2*(10*b^11*d^3 - 10*a^3*b^8*e^3 + 30*a^2*b^9*d*e^2 - 30*a*b^10*d^2*e) + b^11*(d + e*x)^
5 - (5*b^11*d - 5*a*b^10*e)*(d + e*x)^4 - b^11*d^5 + (d + e*x)^3*(10*b^11*d^2 + 10*a^2*b^9*e^2 - 20*a*b^10*d*e
) + a^5*b^6*e^5 - 5*a^4*b^7*d*e^4 - 10*a^2*b^9*d^3*e^2 + 10*a^3*b^8*d^2*e^3 + 5*a*b^10*d^4*e) + (2*e^5*(d + e*
x)^(1/2))/b^6 - (693*e^5*atan((b^(1/2)*e^5*(a*e - b*d)^(1/2)*(d + e*x)^(1/2))/(a*e^6 - b*d*e^5))*(a*e - b*d)^(
1/2))/(128*b^(13/2))

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