\(\int \frac {1}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [1674]

Optimal result

Integrand size = 28, antiderivative size = 266 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 \sqrt {d+e x}}+\frac {3003 b^{3/2} e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}} \]

[Out]

-1001/128*e^5/(-a*e+b*d)^6/(e*x+d)^(3/2)-1/5/(-a*e+b*d)/(b*x+a)^5/(e*x+d)^(3/2)+13/40*e/(-a*e+b*d)^2/(b*x+a)^4
/(e*x+d)^(3/2)-143/240*e^2/(-a*e+b*d)^3/(b*x+a)^3/(e*x+d)^(3/2)+429/320*e^3/(-a*e+b*d)^4/(b*x+a)^2/(e*x+d)^(3/
2)-3003/640*e^4/(-a*e+b*d)^5/(b*x+a)/(e*x+d)^(3/2)+3003/128*b^(3/2)*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*
d)^(1/2))/(-a*e+b*d)^(15/2)-3003/128*b*e^5/(-a*e+b*d)^7/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65, 214} \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3003 b^{3/2} e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}}-\frac {3003 b e^5}{128 \sqrt {d+e x} (b d-a e)^7}-\frac {1001 e^5}{128 (d+e x)^{3/2} (b d-a e)^6}-\frac {3003 e^4}{640 (a+b x) (d+e x)^{3/2} (b d-a e)^5}+\frac {429 e^3}{320 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac {143 e^2}{240 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}+\frac {13 e}{40 (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \]

[In]

Int[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-1001*e^5)/(128*(b*d - a*e)^6*(d + e*x)^(3/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(3/2)) + (13*e)/(40*(
b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)) - (143*e^2)/(240*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) + (429*e
^3)/(320*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(3/2)) - (3003*e^4)/(640*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(3/2))
- (3003*b*e^5)/(128*(b*d - a*e)^7*Sqrt[d + e*x]) + (3003*b^(3/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(128*(b*d - a*e)^(15/2))

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 44

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

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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*} \text {integral}& = \int \frac {1}{(a+b x)^6 (d+e x)^{5/2}} \, dx \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {(13 e) \int \frac {1}{(a+b x)^5 (d+e x)^{5/2}} \, dx}{10 (b d-a e)} \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac {\left (143 e^2\right ) \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}} \, dx}{80 (b d-a e)^2} \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac {\left (429 e^3\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{160 (b d-a e)^3} \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {\left (3003 e^4\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{640 (b d-a e)^4} \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {\left (3003 e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^5} \\ & = -\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {\left (3003 b e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^6} \\ & = -\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 \sqrt {d+e x}}-\frac {\left (3003 b^2 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^7} \\ & = -\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 \sqrt {d+e x}}-\frac {\left (3003 b^2 e^4\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 d-a e)^7} \\ & = -\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 \sqrt {d+e x}}+\frac {3003 b^{3/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.02 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {-1280 a^6 e^6+1280 a^5 b e^5 (19 d+13 e x)+5 a^4 b^2 e^4 \left (7119 d^2+38558 d e x+27599 e^2 x^2\right )+10 a^3 b^3 e^3 \left (-2107 d^3+7917 d^2 e x+46475 d e^2 x^2+33891 e^3 x^3\right )+2 a^2 b^4 e^2 \left (5012 d^4-11557 d^3 e x+42042 d^2 e^2 x^2+260403 d e^3 x^3+192192 e^4 x^4\right )+2 a b^5 e \left (-1464 d^5+2704 d^4 e x-6149 d^3 e^2 x^2+21879 d^2 e^3 x^3+141141 d e^4 x^4+105105 e^5 x^5\right )+b^6 \left (384 d^6-624 d^5 e x+1144 d^4 e^2 x^2-2574 d^3 e^3 x^3+9009 d^2 e^4 x^4+60060 d e^5 x^5+45045 e^6 x^6\right )}{(-b d+a e)^7 (a+b x)^5 (d+e x)^{3/2}}+\frac {45045 b^{3/2} e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{15/2}}}{1920} \]

[In]

Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((-1280*a^6*e^6 + 1280*a^5*b*e^5*(19*d + 13*e*x) + 5*a^4*b^2*e^4*(7119*d^2 + 38558*d*e*x + 27599*e^2*x^2) + 10
*a^3*b^3*e^3*(-2107*d^3 + 7917*d^2*e*x + 46475*d*e^2*x^2 + 33891*e^3*x^3) + 2*a^2*b^4*e^2*(5012*d^4 - 11557*d^
3*e*x + 42042*d^2*e^2*x^2 + 260403*d*e^3*x^3 + 192192*e^4*x^4) + 2*a*b^5*e*(-1464*d^5 + 2704*d^4*e*x - 6149*d^
3*e^2*x^2 + 21879*d^2*e^3*x^3 + 141141*d*e^4*x^4 + 105105*e^5*x^5) + b^6*(384*d^6 - 624*d^5*e*x + 1144*d^4*e^2
*x^2 - 2574*d^3*e^3*x^3 + 9009*d^2*e^4*x^4 + 60060*d*e^5*x^5 + 45045*e^6*x^6))/((-(b*d) + a*e)^7*(a + b*x)^5*(
d + e*x)^(3/2)) + (45045*b^(3/2)*e^5*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(15/2)
)/1920

Maple [A] (verified)

Time = 2.41 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(2 e^{5} \left (\frac {b^{2} \left (\frac {\frac {1467 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {9629 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} a^{2} b^{2} e^{2}-\frac {1253}{15} a \,b^{3} d e +\frac {1253}{30} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {12131}{384} a^{3} b \,e^{3}-\frac {12131}{128} a^{2} b^{2} d \,e^{2}+\frac {12131}{128} a \,b^{3} d^{2} e -\frac {12131}{384} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {2373}{256} e^{4} a^{4}-\frac {2373}{64} b \,e^{3} d \,a^{3}+\frac {7119}{128} b^{2} e^{2} d^{2} a^{2}-\frac {2373}{64} a \,b^{3} d^{3} e +\frac {2373}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{7}}-\frac {1}{3 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {3}{2}}}+\frac {6 b}{\left (a e -b d \right )^{7} \sqrt {e x +d}}\right )\)	\(291\)
default	\(2 e^{5} \left (\frac {b^{2} \left (\frac {\frac {1467 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {9629 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} a^{2} b^{2} e^{2}-\frac {1253}{15} a \,b^{3} d e +\frac {1253}{30} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {12131}{384} a^{3} b \,e^{3}-\frac {12131}{128} a^{2} b^{2} d \,e^{2}+\frac {12131}{128} a \,b^{3} d^{2} e -\frac {12131}{384} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {2373}{256} e^{4} a^{4}-\frac {2373}{64} b \,e^{3} d \,a^{3}+\frac {7119}{128} b^{2} e^{2} d^{2} a^{2}-\frac {2373}{64} a \,b^{3} d^{3} e +\frac {2373}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{7}}-\frac {1}{3 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {3}{2}}}+\frac {6 b}{\left (a e -b d \right )^{7} \sqrt {e x +d}}\right )\)	\(291\)
pseudoelliptic	\(-\frac {2 \left (-\frac {9009 b^{2} e^{5} \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256}+\left (\left (-\frac {9009}{256} e^{6} x^{6}-\frac {3}{10} d^{6}-\frac {3003}{64} d \,e^{5} x^{5}-\frac {9009}{1280} d^{2} e^{4} x^{4}+\frac {1287}{640} x^{3} d^{3} e^{3}-\frac {143}{160} d^{4} e^{2} x^{2}+\frac {39}{80} d^{5} e x \right ) b^{6}+\frac {183 \left (-\frac {35035}{488} e^{5} x^{5}-\frac {47047}{488} x^{4} d \,e^{4}-\frac {7293}{488} d^{2} e^{3} x^{3}+\frac {6149}{1464} d^{3} e^{2} x^{2}-\frac {338}{183} d^{4} e x +d^{5}\right ) e a \,b^{5}}{80}-\frac {1253 e^{2} a^{2} \left (\frac {6864}{179} e^{4} x^{4}+\frac {260403}{5012} d \,e^{3} x^{3}+\frac {3003}{358} d^{2} e^{2} x^{2}-\frac {1651}{716} d^{3} e x +d^{4}\right ) b^{4}}{160}+\frac {2107 \left (-\frac {33891}{2107} e^{3} x^{3}-\frac {46475}{2107} d \,e^{2} x^{2}-\frac {1131}{301} d^{2} e x +d^{3}\right ) e^{3} a^{3} b^{3}}{128}-\frac {7119 e^{4} a^{4} \left (\frac {27599}{7119} x^{2} e^{2}+\frac {38558}{7119} d e x +d^{2}\right ) b^{2}}{256}-19 \left (\frac {13 e x}{19}+d \right ) e^{5} a^{5} b +a^{6} e^{6}\right ) \sqrt {\left (a e -b d \right ) b}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} \left (a e -b d \right )^{7}}\)	\(361\)

[In]

int(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*(1/(a*e-b*d)^7*b^2*((1467/256*b^4*(e*x+d)^(9/2)+9629/384*(a*e-b*d)*b^3*(e*x+d)^(7/2)+(1253/30*a^2*b^2*e^
2-1253/15*a*b^3*d*e+1253/30*b^4*d^2)*(e*x+d)^(5/2)+(12131/384*a^3*b*e^3-12131/128*a^2*b^2*d*e^2+12131/128*a*b^
3*d^2*e-12131/384*b^4*d^3)*(e*x+d)^(3/2)+(2373/256*e^4*a^4-2373/64*b*e^3*d*a^3+7119/128*b^2*e^2*d^2*a^2-2373/6
4*a*b^3*d^3*e+2373/256*b^4*d^4)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^5+3003/256/((a*e-b*d)*b)^(1/2)*arctan(b*(e*
x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))-1/3/(a*e-b*d)^6/(e*x+d)^(3/2)+6/(a*e-b*d)^7*b/(e*x+d)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1617 vs. \(2 (226) = 452\).

Time = 0.88 (sec) , antiderivative size = 3244, normalized size of antiderivative = 12.20 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(45045*(b^6*e^7*x^7 + a^5*b*d^2*e^5 + (2*b^6*d*e^6 + 5*a*b^5*e^7)*x^6 + (b^6*d^2*e^5 + 10*a*b^5*d*e^6
+ 10*a^2*b^4*e^7)*x^5 + 5*(a*b^5*d^2*e^5 + 4*a^2*b^4*d*e^6 + 2*a^3*b^3*e^7)*x^4 + 5*(2*a^2*b^4*d^2*e^5 + 4*a^
3*b^3*d*e^6 + a^4*b^2*e^7)*x^3 + (10*a^3*b^3*d^2*e^5 + 10*a^4*b^2*d*e^6 + a^5*b*e^7)*x^2 + (5*a^4*b^2*d^2*e^5
+ 2*a^5*b*d*e^6)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a
*e)))/(b*x + a)) + 2*(45045*b^6*e^6*x^6 + 384*b^6*d^6 - 2928*a*b^5*d^5*e + 10024*a^2*b^4*d^4*e^2 - 21070*a^3*b
^3*d^3*e^3 + 35595*a^4*b^2*d^2*e^4 + 24320*a^5*b*d*e^5 - 1280*a^6*e^6 + 30030*(2*b^6*d*e^5 + 7*a*b^5*e^6)*x^5
+ 3003*(3*b^6*d^2*e^4 + 94*a*b^5*d*e^5 + 128*a^2*b^4*e^6)*x^4 - 858*(3*b^6*d^3*e^3 - 51*a*b^5*d^2*e^4 - 607*a^
2*b^4*d*e^5 - 395*a^3*b^3*e^6)*x^3 + 143*(8*b^6*d^4*e^2 - 86*a*b^5*d^3*e^3 + 588*a^2*b^4*d^2*e^4 + 3250*a^3*b^
3*d*e^5 + 965*a^4*b^2*e^6)*x^2 - 26*(24*b^6*d^5*e - 208*a*b^5*d^4*e^2 + 889*a^2*b^4*d^3*e^3 - 3045*a^3*b^3*d^2
*e^4 - 7415*a^4*b^2*d*e^5 - 640*a^5*b*e^6)*x)*sqrt(e*x + d))/(a^5*b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e
^2 - 35*a^8*b^4*d^6*e^3 + 35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5 + 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d
^7*e^2 - 7*a*b^11*d^6*e^3 + 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7
+ 7*a^6*b^6*d*e^8 - a^7*b^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5
*e^4 - 105*a^4*b^8*d^4*e^5 + 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6
+ (b^12*d^9 + 3*a*b^11*d^8*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^
4*e^5 + 147*a^6*b^6*d^3*e^6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3
*a^2*b^10*d^8*e - 5*a^3*b^9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5
*d^3*e^6 - 15*a^8*b^4*d^2*e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e
+ 15*a^4*b^8*d^7*e^2 + 7*a^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^
9*b^3*d^2*e^7 + 3*a^10*b^2*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2
- 147*a^6*b^6*d^6*e^3 + 21*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 - 105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 -
3*a^11*b*d*e^8 - a^12*e^9)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3
+ 105*a^8*b^4*d^5*e^4 - 35*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 + 9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*x), 1/1920
*(45045*(b^6*e^7*x^7 + a^5*b*d^2*e^5 + (2*b^6*d*e^6 + 5*a*b^5*e^7)*x^6 + (b^6*d^2*e^5 + 10*a*b^5*d*e^6 + 10*a^
2*b^4*e^7)*x^5 + 5*(a*b^5*d^2*e^5 + 4*a^2*b^4*d*e^6 + 2*a^3*b^3*e^7)*x^4 + 5*(2*a^2*b^4*d^2*e^5 + 4*a^3*b^3*d*
e^6 + a^4*b^2*e^7)*x^3 + (10*a^3*b^3*d^2*e^5 + 10*a^4*b^2*d*e^6 + a^5*b*e^7)*x^2 + (5*a^4*b^2*d^2*e^5 + 2*a^5*
b*d*e^6)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (4504
5*b^6*e^6*x^6 + 384*b^6*d^6 - 2928*a*b^5*d^5*e + 10024*a^2*b^4*d^4*e^2 - 21070*a^3*b^3*d^3*e^3 + 35595*a^4*b^2
*d^2*e^4 + 24320*a^5*b*d*e^5 - 1280*a^6*e^6 + 30030*(2*b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 3003*(3*b^6*d^2*e^4 + 94
*a*b^5*d*e^5 + 128*a^2*b^4*e^6)*x^4 - 858*(3*b^6*d^3*e^3 - 51*a*b^5*d^2*e^4 - 607*a^2*b^4*d*e^5 - 395*a^3*b^3*
e^6)*x^3 + 143*(8*b^6*d^4*e^2 - 86*a*b^5*d^3*e^3 + 588*a^2*b^4*d^2*e^4 + 3250*a^3*b^3*d*e^5 + 965*a^4*b^2*e^6)
*x^2 - 26*(24*b^6*d^5*e - 208*a*b^5*d^4*e^2 + 889*a^2*b^4*d^3*e^3 - 3045*a^3*b^3*d^2*e^4 - 7415*a^4*b^2*d*e^5
- 640*a^5*b*e^6)*x)*sqrt(e*x + d))/(a^5*b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e^2 - 35*a^8*b^4*d^6*e^3 +
35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5 + 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d^7*e^2 - 7*a*b^11*d^6*e^3
+ 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7 + 7*a^6*b^6*d*e^8 - a^7*b
^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5*e^4 - 105*a^4*b^8*d^4*e^5
+ 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6 + (b^12*d^9 + 3*a*b^11*d^8
*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^4*e^5 + 147*a^6*b^6*d^3*e^
6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3*a^2*b^10*d^8*e - 5*a^3*b^
9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5*d^3*e^6 - 15*a^8*b^4*d^2*
e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e + 15*a^4*b^8*d^7*e^2 + 7*a
^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^9*b^3*d^2*e^7 + 3*a^10*b^2
*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2 - 147*a^6*b^6*d^6*e^3 + 21
*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 - 105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 - 3*a^11*b*d*e^8 - a^12*e^9
)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3 + 105*a^8*b^4*d^5*e^4 - 3
5*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 + 9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^(5/2)/(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(a*e-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (226) = 452\).

Time = 0.30 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.44 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3003 \, b^{2} e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (18 \, {\left (e x + d\right )} b e^{5} + b d e^{5} - a e^{6}\right )}}{3 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} - \frac {22005 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{6} e^{5} - 96290 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} d e^{5} + 160384 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d^{2} e^{5} - 121310 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{3} e^{5} + 35595 \, \sqrt {e x + d} b^{6} d^{4} e^{5} + 96290 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{5} e^{6} - 320768 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} d e^{6} + 363930 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d^{2} e^{6} - 142380 \, \sqrt {e x + d} a b^{5} d^{3} e^{6} + 160384 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{4} e^{7} - 363930 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} d e^{7} + 213570 \, \sqrt {e x + d} a^{2} b^{4} d^{2} e^{7} + 121310 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{3} e^{8} - 142380 \, \sqrt {e x + d} a^{3} b^{3} d e^{8} + 35595 \, \sqrt {e x + d} a^{4} b^{2} e^{9}}{1920 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]

[In]

integrate(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-3003/128*b^2*e^5*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2
- 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*sqrt(-b^2*d + a*b*e)
) - 2/3*(18*(e*x + d)*b*e^5 + b*d*e^5 - a*e^6)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4
*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(e*x + d)^(3/2)) - 1/1920*(22005*(e*
x + d)^(9/2)*b^6*e^5 - 96290*(e*x + d)^(7/2)*b^6*d*e^5 + 160384*(e*x + d)^(5/2)*b^6*d^2*e^5 - 121310*(e*x + d)
^(3/2)*b^6*d^3*e^5 + 35595*sqrt(e*x + d)*b^6*d^4*e^5 + 96290*(e*x + d)^(7/2)*a*b^5*e^6 - 320768*(e*x + d)^(5/2
)*a*b^5*d*e^6 + 363930*(e*x + d)^(3/2)*a*b^5*d^2*e^6 - 142380*sqrt(e*x + d)*a*b^5*d^3*e^6 + 160384*(e*x + d)^(
5/2)*a^2*b^4*e^7 - 363930*(e*x + d)^(3/2)*a^2*b^4*d*e^7 + 213570*sqrt(e*x + d)*a^2*b^4*d^2*e^7 + 121310*(e*x +
d)^(3/2)*a^3*b^3*e^8 - 142380*sqrt(e*x + d)*a^3*b^3*d*e^8 + 35595*sqrt(e*x + d)*a^4*b^2*e^9)/((b^7*d^7 - 7*a*
b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6
- a^7*e^7)*((e*x + d)*b - b*d + a*e)^5)

Mupad [B] (verification not implemented)

Time = 10.32 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.09 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {27599\,b^2\,e^5\,{\left (d+e\,x\right )}^2}{384\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,e^5}{3\,\left (a\,e-b\,d\right )}+\frac {11297\,b^3\,e^5\,{\left (d+e\,x\right )}^3}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {1001\,b^4\,e^5\,{\left (d+e\,x\right )}^4}{5\,{\left (a\,e-b\,d\right )}^5}+\frac {7007\,b^5\,e^5\,{\left (d+e\,x\right )}^5}{64\,{\left (a\,e-b\,d\right )}^6}+\frac {3003\,b^6\,e^5\,{\left (d+e\,x\right )}^6}{128\,{\left (a\,e-b\,d\right )}^7}+\frac {26\,b\,e^5\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}}{{\left (d+e\,x\right )}^{3/2}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{7/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{5/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{13/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{11/2}+{\left (d+e\,x\right )}^{9/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}+\frac {3003\,b^{3/2}\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}{{\left (a\,e-b\,d\right )}^{15/2}}\right )}{128\,{\left (a\,e-b\,d\right )}^{15/2}} \]

[In]

int(1/((d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

((27599*b^2*e^5*(d + e*x)^2)/(384*(a*e - b*d)^3) - (2*e^5)/(3*(a*e - b*d)) + (11297*b^3*e^5*(d + e*x)^3)/(64*(
a*e - b*d)^4) + (1001*b^4*e^5*(d + e*x)^4)/(5*(a*e - b*d)^5) + (7007*b^5*e^5*(d + e*x)^5)/(64*(a*e - b*d)^6) +
(3003*b^6*e^5*(d + e*x)^6)/(128*(a*e - b*d)^7) + (26*b*e^5*(d + e*x))/(3*(a*e - b*d)^2))/((d + e*x)^(3/2)*(a^
5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) - (d + e*x)^(7/2)*(
10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^4*d^2*e) + (d + e*x)^(5/2)*(5*b^5*d^4 + 5*a^4*b*e^4 -
20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) + b^5*(d + e*x)^(13/2) - (5*b^5*d - 5*a*b^4*e)*(d + e*
x)^(11/2) + (d + e*x)^(9/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e)) + (3003*b^(3/2)*e^5*atan((b^(1/2)*(d
+ e*x)^(1/2)*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d
^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6))/(a*e - b*d)^(15/2)))/(128*(a*e - b*d)^(15/2))