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

Optimal. Leaf size=279 \[ \frac {4}{25} (2 x+1)^{5/2}+\frac {16}{75} (2 x+1)^{3/2}-\frac {76}{125} \sqrt {2 x+1}-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

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Rubi [A]  time = 0.61, antiderivative size = 279, 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 = {703, 824, 826, 1169, 634, 618, 204, 628} \begin {gather*} \frac {4}{25} (2 x+1)^{5/2}+\frac {16}{75} (2 x+1)^{3/2}-\frac {76}{125} \sqrt {2 x+1}-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\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}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\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),x]

[Out]

(-76*Sqrt[1 + 2*x])/125 + (16*(1 + 2*x)^(3/2))/75 + (4*(1 + 2*x)^(5/2))/25 + (Sqrt[(2*(-168698 + 42875*Sqrt[35
]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/125 - (Sqrt[(2*(-16869
8 + 42875*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/125 -
(Sqrt[(168698 + 42875*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/125
+ (Sqrt[(168698 + 42875*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/1
25

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 703

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1))/(c*
(m - 1)), x] + Dist[1/c, Int[((d + e*x)^(m - 2)*Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x])/(a + b*x + c*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] && NeQ[2*c*d - b*
e, 0] && GtQ[m, 1]

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}}{2+3 x+5 x^2} \, dx &=\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{5} \int \frac {(1+2 x)^{3/2} (-3+8 x)}{2+3 x+5 x^2} \, dx\\ &=\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{25} \int \frac {(-47-38 x) \sqrt {1+2 x}}{2+3 x+5 x^2} \, dx\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \int \frac {-83-432 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {2}{125} \operatorname {Subst}\left (\int \frac {266-432 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {\operatorname {Subst}\left (\int \frac {266 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (266+432 \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 )}{125 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {266 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (266+432 \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 )}{125 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{625} \left (-216+19 \sqrt {35}\right ) \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 )+\frac {1}{625} \left (-216+19 \sqrt {35}\right ) \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 )-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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 {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{625} \left (2 \left (216-19 \sqrt {35}\right )\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 )+\frac {1}{625} \left (2 \left (216-19 \sqrt {35}\right )\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 )\\ &=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \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}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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 = 0.31, size = 134, normalized size = 0.48 \begin {gather*} \frac {2 \left (620 \sqrt {2 x+1} \left (30 x^2+50 x-11\right )+3 \sqrt {10-5 i \sqrt {31}} \left (589+178 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+3 \sqrt {10+5 i \sqrt {31}} \left (589-178 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right )}{58125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(620*Sqrt[1 + 2*x]*(-11 + 50*x + 30*x^2) + 3*Sqrt[10 - (5*I)*Sqrt[31]]*(589 + (178*I)*Sqrt[31])*ArcTanh[Sqr
t[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 3*Sqrt[10 + (5*I)*Sqrt[31]]*(589 - (178*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x
]/Sqrt[2 + I*Sqrt[31]]]))/58125

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IntegrateAlgebraic [C]  time = 0.74, size = 138, normalized size = 0.49 \begin {gather*} \frac {4}{375} \sqrt {2 x+1} \left (15 (2 x+1)^2+20 (2 x+1)-57\right )-\frac {2}{125} \sqrt {\frac {1}{155} \left (-168698-34021 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}-\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )-\frac {2}{125} \sqrt {\frac {1}{155} \left (-168698+34021 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),x]

[Out]

(4*Sqrt[1 + 2*x]*(-57 + 20*(1 + 2*x) + 15*(1 + 2*x)^2))/375 - (2*Sqrt[(-168698 - (34021*I)*Sqrt[31])/155]*ArcT
an[Sqrt[-2/7 - (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/125 - (2*Sqrt[(-168698 + (34021*I)*Sqrt[31])/155]*ArcTan[Sqrt[-
2/7 + (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/125

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fricas [B]  time = 0.45, size = 468, normalized size = 1.68 \begin {gather*} \frac {1}{1752203762968750} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (168698 \, \sqrt {35} \sqrt {31} + 1500625 \, \sqrt {31}\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} \log \left (\frac {26582500}{34021} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (216 \, \sqrt {35} \sqrt {31} + 665 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + 353314653125000 \, x + 35331465312500 \, \sqrt {35} + 176657326562500\right ) - \frac {1}{1752203762968750} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (168698 \, \sqrt {35} \sqrt {31} + 1500625 \, \sqrt {31}\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} \log \left (-\frac {26582500}{34021} \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (216 \, \sqrt {35} \sqrt {31} + 665 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + 353314653125000 \, x + 35331465312500 \, \sqrt {35} + 176657326562500\right ) + \frac {2}{830703125} \cdot 42875^{\frac {1}{4}} \sqrt {155} \sqrt {35} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} \arctan \left (\frac {1}{2044682140744622640625} \cdot 42875^{\frac {3}{4}} \sqrt {34021} \sqrt {217} \sqrt {155} \sqrt {42875^{\frac {1}{4}} \sqrt {155} {\left (216 \, \sqrt {35} \sqrt {31} + 665 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + 452181616250 \, x + 45218161625 \, \sqrt {35} + 226090808125} {\left (19 \, \sqrt {35} + 216\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} - \frac {1}{1582635656875} \cdot 42875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (19 \, \sqrt {35} + 216\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + \frac {2}{830703125} \cdot 42875^{\frac {1}{4}} \sqrt {155} \sqrt {35} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} \arctan \left (\frac {1}{715638749260617924218750} \cdot 42875^{\frac {3}{4}} \sqrt {34021} \sqrt {155} \sqrt {-26582500 \cdot 42875^{\frac {1}{4}} \sqrt {155} {\left (216 \, \sqrt {35} \sqrt {31} + 665 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + 12020117813965625000 \, x + 1202011781396562500 \, \sqrt {35} + 6010058906982812500} {\left (19 \, \sqrt {35} + 216\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} - \frac {1}{1582635656875} \cdot 42875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (19 \, \sqrt {35} + 216\right )} \sqrt {-14465853500 \, \sqrt {35} + 128678593750} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + \frac {8}{375} \, {\left (30 \, x^{2} + 50 \, x - 11\right )} \sqrt {2 \, x + 1} \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),x, algorithm="fricas")

[Out]

1/1752203762968750*42875^(1/4)*sqrt(155)*(168698*sqrt(35)*sqrt(31) + 1500625*sqrt(31))*sqrt(-14465853500*sqrt(
35) + 128678593750)*log(26582500/34021*42875^(1/4)*sqrt(155)*(216*sqrt(35)*sqrt(31) + 665*sqrt(31))*sqrt(2*x +
1)*sqrt(-14465853500*sqrt(35) + 128678593750) + 353314653125000*x + 35331465312500*sqrt(35) + 176657326562500
) - 1/1752203762968750*42875^(1/4)*sqrt(155)*(168698*sqrt(35)*sqrt(31) + 1500625*sqrt(31))*sqrt(-14465853500*s
qrt(35) + 128678593750)*log(-26582500/34021*42875^(1/4)*sqrt(155)*(216*sqrt(35)*sqrt(31) + 665*sqrt(31))*sqrt(
2*x + 1)*sqrt(-14465853500*sqrt(35) + 128678593750) + 353314653125000*x + 35331465312500*sqrt(35) + 1766573265
62500) + 2/830703125*42875^(1/4)*sqrt(155)*sqrt(35)*sqrt(-14465853500*sqrt(35) + 128678593750)*arctan(1/204468
2140744622640625*42875^(3/4)*sqrt(34021)*sqrt(217)*sqrt(155)*sqrt(42875^(1/4)*sqrt(155)*(216*sqrt(35)*sqrt(31)
+ 665*sqrt(31))*sqrt(2*x + 1)*sqrt(-14465853500*sqrt(35) + 128678593750) + 452181616250*x + 45218161625*sqrt(
35) + 226090808125)*(19*sqrt(35) + 216)*sqrt(-14465853500*sqrt(35) + 128678593750) - 1/1582635656875*42875^(3/
4)*sqrt(155)*sqrt(2*x + 1)*(19*sqrt(35) + 216)*sqrt(-14465853500*sqrt(35) + 128678593750) - 1/31*sqrt(35)*sqrt
(31) - 2/31*sqrt(31)) + 2/830703125*42875^(1/4)*sqrt(155)*sqrt(35)*sqrt(-14465853500*sqrt(35) + 128678593750)*
arctan(1/715638749260617924218750*42875^(3/4)*sqrt(34021)*sqrt(155)*sqrt(-26582500*42875^(1/4)*sqrt(155)*(216*
sqrt(35)*sqrt(31) + 665*sqrt(31))*sqrt(2*x + 1)*sqrt(-14465853500*sqrt(35) + 128678593750) + 12020117813965625
000*x + 1202011781396562500*sqrt(35) + 6010058906982812500)*(19*sqrt(35) + 216)*sqrt(-14465853500*sqrt(35) + 1
28678593750) - 1/1582635656875*42875^(3/4)*sqrt(155)*sqrt(2*x + 1)*(19*sqrt(35) + 216)*sqrt(-14465853500*sqrt(
35) + 128678593750) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 8/375*(30*x^2 + 50*x - 11)*sqrt(2*x + 1)

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giac [B]  time = 1.22, size = 614, normalized size = 2.20

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),x, algorithm="giac")

[Out]

4/25*(2*x + 1)^(5/2) - 1/1162984375*sqrt(31)*(11340*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35)
+ 2450) - 54*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 108*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) +
22680*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 162925*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35
) + 2450) - 325850*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(3
5) + 1/2) + sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 1/1162984375*sqrt(31)*(11340*sqrt(31)*(7/5)^(3/4)*(2*
sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 54*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 108*(7/5)^(
3/4)*(140*sqrt(35) + 2450)^(3/2) + 22680*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 162925*sqrt
(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 325850*(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)) - 1/2325968750*sqrt(
31)*(54*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 11340*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450
)*(2*sqrt(35) - 35) - 22680*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 108*(7/5)^(3/4)*(-140*s
qrt(35) + 2450)^(3/2) - 162925*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 325850*(7/5)^(1/4)*sqrt(-140*s
qrt(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) + 1/23259687
50*sqrt(31)*(54*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 11340*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35
) + 2450)*(2*sqrt(35) - 35) - 22680*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 108*(7/5)^(3/4)
*(-140*sqrt(35) + 2450)^(3/2) - 162925*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 325850*(7/5)^(1/4)*sqr
t(-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
6/75*(2*x + 1)^(3/2) - 76/125*sqrt(2*x + 1)

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maple [B]  time = 1.99, size = 634, normalized size = 2.27 \begin {gather*} -\frac {178 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {233 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {76 \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 )}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {178 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {233 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {76 \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 )}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {89 \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 )}{3875}-\frac {233 \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 )}{38750}+\frac {89 \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 )}{3875}+\frac {233 \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 )}{38750}+\frac {4 \left (2 x +1\right )^{\frac {5}{2}}}{25}+\frac {16 \left (2 x +1\right )^{\frac {3}{2}}}{75}-\frac {76 \sqrt {2 x +1}}{125} \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),x)

[Out]

4/25*(2*x+1)^(5/2)+16/75*(2*x+1)^(3/2)-76/125*(2*x+1)^(1/2)+89/3875*ln(5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(
1/2)*5^(1/2)*(2*x+1)^(1/2)+10*x+5)*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+233/38750*ln(5^(1/2)*7^(1/2)+(2*5^(1/2)
*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+10*x+5)*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-178/3875/(10*5^(1/2)*7^(1/
2)-20)^(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))*5^(1
/2)*(2*5^(1/2)*7^(1/2)+4)*7^(1/2)-233/3875/(10*5^(1/2)*7^(1/2)-20)^(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))*(2*5^(1/2)*7^(1/2)+4)+76/125/(10*5^(1/2)*7^(1/2)-20)^(
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))*5^(1/2)*7^(1
/2)-89/3875*ln(5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+10*x+5)*7^(1/2)*(2*5^(1/2)*7^
(1/2)+4)^(1/2)-233/38750*ln(5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+10*x+5)*5^(1/2)*
(2*5^(1/2)*7^(1/2)+4)^(1/2)-178/3875/(10*5^(1/2)*7^(1/2)-20)^(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))*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)*7^(1/2)-233/3875/(10*5^(1/2)*
7^(1/2)-20)^(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)
)*(2*5^(1/2)*7^(1/2)+4)+76/125/(10*5^(1/2)*7^(1/2)-20)^(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))*5^(1/2)*7^(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}}}{5 \, x^{2} + 3 \, x + 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),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.04, size = 200, normalized size = 0.72 \begin {gather*} \frac {16\,{\left (2\,x+1\right )}^{3/2}}{75}-\frac {76\,\sqrt {2\,x+1}}{125}+\frac {4\,{\left (2\,x+1\right )}^{5/2}}{25}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}+\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}-\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375} \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),x)

[Out]

(16*(2*x + 1)^(3/2))/75 - (76*(2*x + 1)^(1/2))/125 + (4*(2*x + 1)^(5/2))/25 + (155^(1/2)*atan((155^(1/2)*(31^(
1/2)*34021i + 168698)^(1/2)*(2*x + 1)^(1/2)*4354688i)/(6103515625*((31^(1/2)*579173504i)/1220703125 - 54259412
48/1220703125)) + (8709376*31^(1/2)*155^(1/2)*(31^(1/2)*34021i + 168698)^(1/2)*(2*x + 1)^(1/2))/(189208984375*
((31^(1/2)*579173504i)/1220703125 - 5425941248/1220703125)))*(31^(1/2)*34021i + 168698)^(1/2)*2i)/19375 - (155
^(1/2)*atan((155^(1/2)*(168698 - 31^(1/2)*34021i)^(1/2)*(2*x + 1)^(1/2)*4354688i)/(6103515625*((31^(1/2)*57917
3504i)/1220703125 + 5425941248/1220703125)) - (8709376*31^(1/2)*155^(1/2)*(168698 - 31^(1/2)*34021i)^(1/2)*(2*
x + 1)^(1/2))/(189208984375*((31^(1/2)*579173504i)/1220703125 + 5425941248/1220703125)))*(168698 - 31^(1/2)*34
021i)^(1/2)*2i)/19375

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sympy [A]  time = 87.70, size = 292, normalized size = 1.05 \begin {gather*} \frac {4 \left (2 x + 1\right )^{\frac {5}{2}}}{25} + \frac {16 \left (2 x + 1\right )^{\frac {3}{2}}}{75} - \frac {76 \sqrt {2 x + 1}}{125} + 4 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left (t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )} - \frac {408 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left (t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {84 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left (t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} - \frac {12 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left (t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {112 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left (t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{125} + \frac {504 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left (t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} + \frac {976 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left (t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{125} \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),x)

[Out]

4*(2*x + 1)**(5/2)/25 + 16*(2*x + 1)**(3/2)/75 - 76*sqrt(2*x + 1)/125 + 4*RootSum(1230080*_t**4 + 1984*_t**2 +
7, Lambda(_t, _t*log(9920*_t**3 + 8*_t + sqrt(2*x + 1)))) - 408*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambd
a(_t, _t*log(9920*_t**3 + 8*_t + sqrt(2*x + 1))))/25 - 84*RootSum(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _
t*log(-27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1))))/5 - 12*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _
t*log(9920*_t**3 + 8*_t + sqrt(2*x + 1))))/5 + 112*RootSum(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-
27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1))))/125 + 504*RootSum(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*lo
g(-27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1))))/25 + 976*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*
log(9920*_t**3 + 8*_t + sqrt(2*x + 1))))/125

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