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

Optimal. Leaf size=300 \[ -\frac {(5-4 x) (2 x+1)^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac {3 (78 x+11) \sqrt {2 x+1}}{1922 \left (5 x^2+3 x+2\right )}+\frac {3 \sqrt {\frac {1}{310} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}-\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.44, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {738, 820, 826, 1169, 634, 618, 204, 628} \begin {gather*} -\frac {(5-4 x) (2 x+1)^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac {3 (78 x+11) \sqrt {2 x+1}}{1922 \left (5 x^2+3 x+2\right )}+\frac {3 \sqrt {\frac {1}{310} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}-\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\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 {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\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)^(5/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

-((5 - 4*x)*(1 + 2*x)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*Sqrt[1 + 2*x]*(11 + 78*x))/(1922*(2 + 3*x + 5*x^2))
- (3*Sqrt[(15082 + 2705*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt
[35])]])/961 + (3*Sqrt[(15082 + 2705*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[1
0*(-2 + Sqrt[35])]])/961 + (3*Sqrt[(-15082 + 2705*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1
+ 2*x] + 5*(1 + 2*x)])/1922 - (3*Sqrt[(-15082 + 2705*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sq
rt[1 + 2*x] + 5*(1 + 2*x)])/1922

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 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], 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] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

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)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {\sqrt {1+2 x} (27+12 x)}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {201+234 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{1922}\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {1}{961} \operatorname {Subst}\left (\int \frac {168+234 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {168 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (168-234 \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 )}{1922 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {168 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (168-234 \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 )}{1922 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (39+4 \sqrt {35}\right )\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 )}{9610}+\frac {\left (3 \left (39+4 \sqrt {35}\right )\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 )}{9610}+\frac {\left (3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )}\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 )}{1922}-\frac {\left (3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )}\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 )}{1922}\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {\left (3 \left (39+4 \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 )}{4805}-\frac {\left (3 \left (39+4 \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 )}{4805}\\ &=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {7541}{155}+\frac {541 \sqrt {35}}{62}} \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 {3}{961} \sqrt {\frac {7541}{155}+\frac {541 \sqrt {35}}{62}} \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 {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.58, size = 209, normalized size = 0.70 \begin {gather*} \frac {\frac {(480 x+1973) (2 x+1)^{7/2}}{5 x^2+3 x+2}+\frac {217 (20 x+37) (2 x+1)^{7/2}}{\left (5 x^2+3 x+2\right )^2}-192 (2 x+1)^{5/2}-1540 (2 x+1)^{3/2}-2352 \sqrt {2 x+1}+\frac {294 \left (\sqrt {2-i \sqrt {31}} \left (124-47 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+\sqrt {2+i \sqrt {31}} \left (124+47 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right )}{31 \sqrt {5}}}{94178} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2352*Sqrt[1 + 2*x] - 1540*(1 + 2*x)^(3/2) - 192*(1 + 2*x)^(5/2) + (217*(1 + 2*x)^(7/2)*(37 + 20*x))/(2 + 3*x
+ 5*x^2)^2 + ((1 + 2*x)^(7/2)*(1973 + 480*x))/(2 + 3*x + 5*x^2) + (294*(Sqrt[2 - I*Sqrt[31]]*(124 - (47*I)*Sq
rt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + Sqrt[2 + I*Sqrt[31]]*(124 + (47*I)*Sqrt[31])*ArcTanh[Sq
rt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/(31*Sqrt[5]))/94178

________________________________________________________________________________________

IntegrateAlgebraic [C]  time = 2.03, size = 167, normalized size = 0.56 \begin {gather*} \frac {2 \sqrt {2 x+1} \left (585 (2 x+1)^3-640 (2 x+1)^2+287 (2 x+1)-588\right )}{961 \left (5 (2 x+1)^2-4 (2 x+1)+7\right )^2}+\frac {3}{961} \sqrt {\frac {1}{155} \left (15082-961 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}-\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )+\frac {3}{961} \sqrt {\frac {1}{155} \left (15082+961 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)^(5/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

(2*Sqrt[1 + 2*x]*(-588 + 287*(1 + 2*x) - 640*(1 + 2*x)^2 + 585*(1 + 2*x)^3))/(961*(7 - 4*(1 + 2*x) + 5*(1 + 2*
x)^2)^2) + (3*Sqrt[(15082 - (961*I)*Sqrt[31])/155]*ArcTan[Sqrt[-2/7 - (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/961 + (3
*Sqrt[(15082 + (961*I)*Sqrt[31])/155]*ArcTan[Sqrt[-2/7 + (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/961

________________________________________________________________________________________

fricas [B]  time = 0.45, size = 617, normalized size = 2.06 \begin {gather*} \frac {357492 \cdot 256095875^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \arctan \left (\frac {1}{694138872776299934375} \cdot 256095875^{\frac {3}{4}} \sqrt {2705} \sqrt {217} \sqrt {155} \sqrt {256095875^{\frac {1}{4}} \sqrt {155} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 28204629250 \, x + 2820462925 \, \sqrt {35} + 14102314625} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (4 \, \sqrt {35} - 39\right )} - \frac {1}{7629352212125} \cdot 256095875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (4 \, \sqrt {35} - 39\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 357492 \cdot 256095875^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \arctan \left (\frac {1}{2082416618328899803125} \cdot 256095875^{\frac {3}{4}} \sqrt {217} \sqrt {155} \sqrt {-24345 \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 686641699091250 \, x + 68664169909125 \, \sqrt {35} + 343320849545625} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (4 \, \sqrt {35} - 39\right )} - \frac {1}{7629352212125} \cdot 256095875^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (4 \, \sqrt {35} - 39\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 3 \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (15082 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 94675 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \log \left (\frac {24345}{217} \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 3164247461250 \, x + 316424746125 \, \sqrt {35} + 1582123730625\right ) - 3 \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (15082 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 94675 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \log \left (-\frac {24345}{217} \cdot 256095875^{\frac {1}{4}} \sqrt {155} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 3164247461250 \, x + 316424746125 \, \sqrt {35} + 1582123730625\right ) + 874343506750 \, {\left (1170 \, x^{3} + 1115 \, x^{2} + 381 \, x - 89\right )} \sqrt {2 \, x + 1}}{1680488219973500 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680488219973500*(357492*256095875^(1/4)*sqrt(155)*sqrt(35)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(81593
620*sqrt(35) + 512191750)*arctan(1/694138872776299934375*256095875^(3/4)*sqrt(2705)*sqrt(217)*sqrt(155)*sqrt(2
56095875^(1/4)*sqrt(155)*(39*sqrt(35)*sqrt(31) - 140*sqrt(31))*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 51219175
0) + 28204629250*x + 2820462925*sqrt(35) + 14102314625)*sqrt(81593620*sqrt(35) + 512191750)*(4*sqrt(35) - 39)
- 1/7629352212125*256095875^(3/4)*sqrt(155)*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 512191750)*(4*sqrt(35) - 39
) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 357492*256095875^(1/4)*sqrt(155)*sqrt(35)*(25*x^4 + 30*x^3 + 29*
x^2 + 12*x + 4)*sqrt(81593620*sqrt(35) + 512191750)*arctan(1/2082416618328899803125*256095875^(3/4)*sqrt(217)*
sqrt(155)*sqrt(-24345*256095875^(1/4)*sqrt(155)*(39*sqrt(35)*sqrt(31) - 140*sqrt(31))*sqrt(2*x + 1)*sqrt(81593
620*sqrt(35) + 512191750) + 686641699091250*x + 68664169909125*sqrt(35) + 343320849545625)*sqrt(81593620*sqrt(
35) + 512191750)*(4*sqrt(35) - 39) - 1/7629352212125*256095875^(3/4)*sqrt(155)*sqrt(2*x + 1)*sqrt(81593620*sqr
t(35) + 512191750)*(4*sqrt(35) - 39) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 3*256095875^(1/4)*sqrt(155)*(
15082*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 94675*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*
x + 4))*sqrt(81593620*sqrt(35) + 512191750)*log(24345/217*256095875^(1/4)*sqrt(155)*(39*sqrt(35)*sqrt(31) - 14
0*sqrt(31))*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 512191750) + 3164247461250*x + 316424746125*sqrt(35) + 1582
123730625) - 3*256095875^(1/4)*sqrt(155)*(15082*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 9467
5*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt(81593620*sqrt(35) + 512191750)*log(-24345/217*256095875
^(1/4)*sqrt(155)*(39*sqrt(35)*sqrt(31) - 140*sqrt(31))*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 512191750) + 316
4247461250*x + 316424746125*sqrt(35) + 1582123730625) + 874343506750*(1170*x^3 + 1115*x^2 + 381*x - 89)*sqrt(2
*x + 1))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

________________________________________________________________________________________

giac [B]  time = 1.43, size = 642, normalized size = 2.14

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/71528191000*sqrt(31)*(8190*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 39*sqrt(31)*(
7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 78*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 16380*(7/5)^(3/4)*sqrt(
140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 274400*(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)) + 3/71528191000*sqrt(31)*(8190*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140
*sqrt(35) + 2450) - 39*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 78*(7/5)^(3/4)*(140*sqrt(35) + 2450
)^(3/2) + 16380*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(-14
0*sqrt(35) + 2450) + 274400*(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)) + 3/143056382000*sqrt(31)*(39*sqrt(31)*(7/5)^
(3/4)*(140*sqrt(35) + 2450)^(3/2) + 8190*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 16
380*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 78*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 1
37200*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 274400*(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) - 3/143056382000*sqrt(31)*(39*sqrt(31
)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 8190*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) -
35) - 16380*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 78*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(
3/2) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 274400*(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) + 2/961*(585*(2*x + 1)^(7/2)
- 640*(2*x + 1)^(5/2) + 287*(2*x + 1)^(3/2) - 588*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)^2

________________________________________________________________________________________

maple [B]  time = 0.19, size = 662, normalized size = 2.21 \begin {gather*} -\frac {327 \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 )}{29791 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {141 \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 )}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {24 \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 )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {327 \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 )}{29791 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {141 \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 )}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {24 \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 )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {327 \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 )}{297910}+\frac {141 \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 )}{119164}+\frac {327 \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 )}{297910}-\frac {141 \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 )}{119164}+\frac {\frac {1170 \left (2 x +1\right )^{\frac {7}{2}}}{961}-\frac {1280 \left (2 x +1\right )^{\frac {5}{2}}}{961}+\frac {574 \left (2 x +1\right )^{\frac {3}{2}}}{961}-\frac {1176 \sqrt {2 x +1}}{961}}{\left (-8 x +5 \left (2 x +1\right )^{2}+3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1600*(117/153760*(2*x+1)^(7/2)-4/4805*(2*x+1)^(5/2)+287/768800*(2*x+1)^(3/2)-147/192200*(2*x+1)^(1/2))/(-8*x+5
*(2*x+1)^2+3)^2+327/297910*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)-141/119164*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)-327/29791/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)*arct
an((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))+141/59582/(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))+24/961/(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))-327/297910*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)+141/119164*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)-327/29
791/(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))+141/59582/(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))+24/961/(
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))/(1
0*5^(1/2)*7^(1/2)-20)^(1/2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.05, size = 245, normalized size = 0.82 \begin {gather*} \frac {\frac {1176\,\sqrt {2\,x+1}}{24025}-\frac {574\,{\left (2\,x+1\right )}^{3/2}}{24025}+\frac {256\,{\left (2\,x+1\right )}^{5/2}}{4805}-\frac {234\,{\left (2\,x+1\right )}^{7/2}}{4805}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{2886003125\,\left (-\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}-\frac {864\,\sqrt {31}\,\sqrt {155}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{89466096875\,\left (-\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}\right )\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{148955}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{2886003125\,\left (\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}+\frac {864\,\sqrt {31}\,\sqrt {155}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{89466096875\,\left (\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}\right )\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{148955} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1176*(2*x + 1)^(1/2))/24025 - (574*(2*x + 1)^(3/2))/24025 + (256*(2*x + 1)^(5/2))/4805 - (234*(2*x + 1)^(7/2
))/4805)/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) - (155^(1/2)*atan((155^(1
/2)*(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(2886003125*((31^(1/2)*12096i)/577200625 - 142128/57
7200625)) - (864*31^(1/2)*155^(1/2)*(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(89466096875*((31^(1/2)*1
2096i)/577200625 - 142128/577200625)))*(- 31^(1/2)*961i - 15082)^(1/2)*3i)/148955 + (155^(1/2)*atan((155^(1/2)
*(31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(2886003125*((31^(1/2)*12096i)/577200625 + 142128/5772006
25)) + (864*31^(1/2)*155^(1/2)*(31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(89466096875*((31^(1/2)*12096i)/
577200625 + 142128/577200625)))*(31^(1/2)*961i - 15082)^(1/2)*3i)/148955

________________________________________________________________________________________

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)**(5/2)/(5*x**2+3*x+2)**3,x)

[Out]

Timed out

________________________________________________________________________________________