3.20.85 \(\int \frac {(1+2 x)^{7/2}}{(2+3 x+5 x^2)^2} \, dx\)

Optimal. Leaf size=296 \[ -\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac {8}{155} (2 x+1)^{3/2}+\frac {604}{775} \sqrt {2 x+1}+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{775} \sqrt {\frac {2}{155} \left (968975 \sqrt {35}-5682718\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{775} \sqrt {\frac {2}{155} \left (968975 \sqrt {35}-5682718\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

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Rubi [A]  time = 0.45, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {738, 824, 826, 1169, 634, 618, 204, 628} \begin {gather*} -\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac {8}{155} (2 x+1)^{3/2}+\frac {604}{775} \sqrt {2 x+1}+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{775} \sqrt {\frac {2}{155} \left (968975 \sqrt {35}-5682718\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{775} \sqrt {\frac {2}{155} \left (968975 \sqrt {35}-5682718\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(604*Sqrt[1 + 2*x])/775 - (8*(1 + 2*x)^(3/2))/155 - ((5 - 4*x)*(1 + 2*x)^(5/2))/(31*(2 + 3*x + 5*x^2)) + (Sqrt
[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[
35])]])/775 - (Sqrt[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/
Sqrt[10*(-2 + Sqrt[35])]])/775 + (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]
*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775 - (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[3
5])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \frac {(29-12 x) (1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx\\ &=-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{155} \int \frac {\sqrt {1+2 x} (193+302 x)}{2+3 x+5 x^2} \, dx\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \int \frac {-243+1628 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {2}{775} \operatorname {Subst}\left (\int \frac {-2114+1628 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-2114 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-2114-1628 \sqrt {\frac {7}{5}}\right ) x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{775 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {-2114 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-2114-1628 \sqrt {\frac {7}{5}}\right ) x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{775 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {1460631-245828 \sqrt {35}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3875}-\frac {\sqrt {1460631-245828 \sqrt {35}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3875}+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \operatorname {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \operatorname {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {\left (2 \sqrt {1460631-245828 \sqrt {35}}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{3875}+\frac {\left (2 \sqrt {1460631-245828 \sqrt {35}}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{3875}\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-2 \sqrt {1+2 x}\right )\right )-\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\right )+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )\\ \end {align*}

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Mathematica [C]  time = 0.44, size = 199, normalized size = 0.67 \begin {gather*} \frac {1}{217} \left (\frac {(20 x+37) (2 x+1)^{9/2}}{5 x^2+3 x+2}-8 (2 x+1)^{7/2}-28 (2 x+1)^{5/2}-\frac {56}{5} (2 x+1)^{3/2}+\frac {4228}{25} \sqrt {2 x+1}-\frac {14 i \left (\sqrt {2-i \sqrt {31}} \left (512 \sqrt {31}-4681 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )-\sqrt {2+i \sqrt {31}} \left (512 \sqrt {31}+4681 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right )}{775 \sqrt {5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

((4228*Sqrt[1 + 2*x])/25 - (56*(1 + 2*x)^(3/2))/5 - 28*(1 + 2*x)^(5/2) - 8*(1 + 2*x)^(7/2) + ((1 + 2*x)^(9/2)*
(37 + 20*x))/(2 + 3*x + 5*x^2) - (((14*I)/775)*(Sqrt[2 - I*Sqrt[31]]*(-4681*I + 512*Sqrt[31])*ArcTanh[Sqrt[5 +
10*x]/Sqrt[2 - I*Sqrt[31]]] - Sqrt[2 + I*Sqrt[31]]*(4681*I + 512*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*
Sqrt[31]]]))/Sqrt[5])/217

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IntegrateAlgebraic [C]  time = 1.66, size = 158, normalized size = 0.53 \begin {gather*} \frac {4 \sqrt {2 x+1} \left (620 (2 x+1)^2-674 (2 x+1)+1057\right )}{775 \left (5 (2 x+1)^2-4 (2 x+1)+7\right )}-\frac {2}{775} \sqrt {\frac {1}{155} \left (-5682718+135439 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}-\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )-\frac {2}{775} \sqrt {\frac {1}{155} \left (-5682718-135439 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}+\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(4*Sqrt[1 + 2*x]*(1057 - 674*(1 + 2*x) + 620*(1 + 2*x)^2))/(775*(7 - 4*(1 + 2*x) + 5*(1 + 2*x)^2)) - (2*Sqrt[(
-5682718 + (135439*I)*Sqrt[31])/155]*ArcTan[Sqrt[-2/7 - (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/775 - (2*Sqrt[(-568271
8 - (135439*I)*Sqrt[31])/155]*ArcTan[Sqrt[-2/7 + (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/775

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fricas [B]  time = 0.45, size = 541, normalized size = 1.83 \begin {gather*} \frac {16794436 \cdot 21898835^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} \arctan \left (\frac {1}{60332699662225359002939375} \cdot 21898835^{\frac {3}{4}} \sqrt {4369} \sqrt {3955} \sqrt {155} \sqrt {21898835^{\frac {1}{4}} \sqrt {155} {\left (814 \, \sqrt {35} \sqrt {31} + 5285 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + 40683471557750 \, x + 4068347155775 \, \sqrt {35} + 20341735778875} {\left (151 \, \sqrt {35} + 814\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} - \frac {1}{3218062600218025} \cdot 21898835^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (151 \, \sqrt {35} + 814\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 16794436 \cdot 21898835^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} \arctan \left (\frac {1}{4223288976355775130205756250} \cdot 21898835^{\frac {3}{4}} \sqrt {4369} \sqrt {155} \sqrt {-19379500 \cdot 21898835^{\frac {1}{4}} \sqrt {155} {\left (814 \, \sqrt {35} \sqrt {31} + 5285 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + 788425337053416125000 \, x + 78842533705341612500 \, \sqrt {35} + 394212668526708062500} {\left (151 \, \sqrt {35} + 814\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} - \frac {1}{3218062600218025} \cdot 21898835^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (151 \, \sqrt {35} + 814\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) - 21898835^{\frac {1}{4}} \sqrt {155} {\left (5682718 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} + 33914125 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} \log \left (\frac {19379500}{4369} \cdot 21898835^{\frac {1}{4}} \sqrt {155} {\left (814 \, \sqrt {35} \sqrt {31} + 5285 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + 180458992230125000 \, x + 18045899223012500 \, \sqrt {35} + 90229496115062500\right ) + 21898835^{\frac {1}{4}} \sqrt {155} {\left (5682718 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} + 33914125 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} \log \left (-\frac {19379500}{4369} \cdot 21898835^{\frac {1}{4}} \sqrt {155} {\left (814 \, \sqrt {35} \sqrt {31} + 5285 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + 180458992230125000 \, x + 18045899223012500 \, \sqrt {35} + 90229496115062500\right ) + 1261187618290250 \, {\left (2480 \, x^{2} + 1132 \, x + 1003\right )} \sqrt {2 \, x + 1}}{977420404174943750 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")

[Out]

1/977420404174943750*(16794436*21898835^(1/4)*sqrt(155)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(-11012823348100*sqrt(3
5) + 65723878543750)*arctan(1/60332699662225359002939375*21898835^(3/4)*sqrt(4369)*sqrt(3955)*sqrt(155)*sqrt(2
1898835^(1/4)*sqrt(155)*(814*sqrt(35)*sqrt(31) + 5285*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) +
65723878543750) + 40683471557750*x + 4068347155775*sqrt(35) + 20341735778875)*(151*sqrt(35) + 814)*sqrt(-11012
823348100*sqrt(35) + 65723878543750) - 1/3218062600218025*21898835^(3/4)*sqrt(155)*sqrt(2*x + 1)*(151*sqrt(35)
+ 814)*sqrt(-11012823348100*sqrt(35) + 65723878543750) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 16794436*2
1898835^(1/4)*sqrt(155)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(-11012823348100*sqrt(35) + 65723878543750)*arctan(1/42
23288976355775130205756250*21898835^(3/4)*sqrt(4369)*sqrt(155)*sqrt(-19379500*21898835^(1/4)*sqrt(155)*(814*sq
rt(35)*sqrt(31) + 5285*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 7884253370534
16125000*x + 78842533705341612500*sqrt(35) + 394212668526708062500)*(151*sqrt(35) + 814)*sqrt(-11012823348100*
sqrt(35) + 65723878543750) - 1/3218062600218025*21898835^(3/4)*sqrt(155)*sqrt(2*x + 1)*(151*sqrt(35) + 814)*sq
rt(-11012823348100*sqrt(35) + 65723878543750) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) - 21898835^(1/4)*sqrt(
155)*(5682718*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) + 33914125*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(-11012823348100*
sqrt(35) + 65723878543750)*log(19379500/4369*21898835^(1/4)*sqrt(155)*(814*sqrt(35)*sqrt(31) + 5285*sqrt(31))*
sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 180458992230125000*x + 18045899223012500*sqrt(
35) + 90229496115062500) + 21898835^(1/4)*sqrt(155)*(5682718*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) + 33914125*sq
rt(31)*(5*x^2 + 3*x + 2))*sqrt(-11012823348100*sqrt(35) + 65723878543750)*log(-19379500/4369*21898835^(1/4)*sq
rt(155)*(814*sqrt(35)*sqrt(31) + 5285*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750)
+ 180458992230125000*x + 18045899223012500*sqrt(35) + 90229496115062500) + 1261187618290250*(2480*x^2 + 1132*x
+ 1003)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)

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giac [B]  time = 1.34, size = 633, normalized size = 2.14

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="giac")

[Out]

1/14421006250*sqrt(31)*(85470*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 407*sqrt(31)
*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 814*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 170940*(7/5)^(3/4)*s
qrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 5179300
*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) + sqrt(2
*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 1/14421006250*sqrt(31)*(85470*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sq
rt(-140*sqrt(35) + 2450) - 407*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 814*(7/5)^(3/4)*(140*sqrt(3
5) + 2450)^(3/2) + 170940*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 2589650*sqrt(31)*(7/5)^(1/
4)*sqrt(-140*sqrt(35) + 2450) - 5179300*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^
(1/4)*sqrt(1/35*sqrt(35) + 1/2) - sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 1/28842012500*sqrt(31)*(407*sqr
t(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 85470*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(3
5) - 35) - 170940*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 814*(7/5)^(3/4)*(-140*sqrt(35) +
2450)^(3/2) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 5179300*(7/5)^(1/4)*sqrt(-140*sqrt(35)
+ 2450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 1/28842012500*sqrt
(31)*(407*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 85470*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 24
50)*(2*sqrt(35) - 35) - 170940*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 814*(7/5)^(3/4)*(-14
0*sqrt(35) + 2450)^(3/2) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 5179300*(7/5)^(1/4)*sqrt(-
140*sqrt(35) + 2450))*log(-2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) + 16/2
5*sqrt(2*x + 1) - 4/775*(178*(2*x + 1)^(3/2) - 189*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)

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maple [B]  time = 0.37, size = 651, normalized size = 2.20 \begin {gather*} \frac {3657 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {512 \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {604 \sqrt {5}\, \sqrt {7}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3657 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {512 \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {604 \sqrt {5}\, \sqrt {7}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{240250}+\frac {256 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{24025}-\frac {3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{240250}-\frac {256 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{24025}+\frac {16 \sqrt {2 x +1}}{25}+\frac {-\frac {712 \left (2 x +1\right )^{\frac {3}{2}}}{3875}+\frac {756 \sqrt {2 x +1}}{3875}}{-\frac {8 x}{5}+\left (2 x +1\right )^{2}+\frac {3}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+1)^(7/2)/(5*x^2+3*x+2)^2,x)

[Out]

16/25*(2*x+1)^(1/2)+16/25*(-89/310*(2*x+1)^(3/2)+189/620*(2*x+1)^(1/2))/((2*x+1)^2-8/5*x+3/5)-3657/240250*5^(1
/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5)-2
56/24025*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+
1)^(1/2)+5)+3657/24025/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)*arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+
4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))+512/24025/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*(2*5
^(1/2)*7^(1/2)+4)*7^(1/2)*arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20
)^(1/2))-604/775/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*7^(1/2)*arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*
(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))+3657/240250*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*
7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5)+256/24025*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(
10*x+5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5)+3657/24025/(10*5^(1/2)*7^(1/2)-20)^(
1/2)*(2*5^(1/2)*7^(1/2)+4)*arctan((-5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-
20)^(1/2))+512/24025/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)*7^(1/2)*arctan((-5^(1/2)*(2*5
^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))-604/775/(10*5^(1/2)*7^(1/2)-20)^(1/2)
*5^(1/2)*7^(1/2)*arctan((-5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x + 1\right )}^{\frac {7}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2, x)

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mupad [B]  time = 0.15, size = 216, normalized size = 0.73 \begin {gather*} \frac {16\,\sqrt {2\,x+1}}{25}+\frac {\frac {756\,\sqrt {2\,x+1}}{3875}-\frac {712\,{\left (2\,x+1\right )}^{3/2}}{3875}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{46923828125\,\left (-\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}+\frac {1118464\,\sqrt {31}\,\sqrt {155}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{1454638671875\,\left (-\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}\right )\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{120125}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{46923828125\,\left (\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}-\frac {1118464\,\sqrt {31}\,\sqrt {155}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{1454638671875\,\left (\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}\right )\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{120125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 1)^(7/2)/(3*x + 5*x^2 + 2)^2,x)

[Out]

(16*(2*x + 1)^(1/2))/25 + ((756*(2*x + 1)^(1/2))/3875 - (712*(2*x + 1)^(3/2))/3875)/((2*x + 1)^2 - (8*x)/5 + 3
/5) - (155^(1/2)*atan((155^(1/2)*(5682718 - 31^(1/2)*135439i)^(1/2)*(2*x + 1)^(1/2)*559232i)/(46923828125*((31
^(1/2)*591108224i)/9384765625 - 2004287488/9384765625)) + (1118464*31^(1/2)*155^(1/2)*(5682718 - 31^(1/2)*1354
39i)^(1/2)*(2*x + 1)^(1/2))/(1454638671875*((31^(1/2)*591108224i)/9384765625 - 2004287488/9384765625)))*(56827
18 - 31^(1/2)*135439i)^(1/2)*2i)/120125 + (155^(1/2)*atan((155^(1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x +
1)^(1/2)*559232i)/(46923828125*((31^(1/2)*591108224i)/9384765625 + 2004287488/9384765625)) - (1118464*31^(1/2
)*155^(1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x + 1)^(1/2))/(1454638671875*((31^(1/2)*591108224i)/93847656
25 + 2004287488/9384765625)))*(31^(1/2)*135439i + 5682718)^(1/2)*2i)/120125

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sympy [F(-1)]  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((1+2*x)**(7/2)/(5*x**2+3*x+2)**2,x)

[Out]

Timed out

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