3.2317 \(\int \frac {(1+2 x)^{5/2}}{(2+3 x+5 x^2)^2} \, dx\)

Optimal. Leaf size=283 \[ -\frac {(5-4 x) (2 x+1)^{3/2}}{31 \left (5 x^2+3 x+2\right )}-\frac {8}{155} \sqrt {2 x+1}+\frac {1}{155} \sqrt {\frac {1}{310} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{155} \sqrt {\frac {1}{310} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

[Out]

-1/31*(5-4*x)*(1+2*x)^(3/2)/(5*x^2+3*x+2)-8/155*(1+2*x)^(1/2)+1/48050*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*
35^(1/2))^(1/2))*(-10130180+3200750*35^(1/2))^(1/2)-1/48050*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^
(1/2))*(-10130180+3200750*35^(1/2))^(1/2)-1/24025*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35
^(1/2))^(1/2))*(10130180+3200750*35^(1/2))^(1/2)+1/24025*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20
+10*35^(1/2))^(1/2))*(10130180+3200750*35^(1/2))^(1/2)

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Rubi [A]  time = 0.41, antiderivative size = 283, 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, 824, 826, 1169, 634, 618, 204, 628} \[ -\frac {(5-4 x) (2 x+1)^{3/2}}{31 \left (5 x^2+3 x+2\right )}-\frac {8}{155} \sqrt {2 x+1}+\frac {1}{155} \sqrt {\frac {1}{310} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{155} \sqrt {\frac {1}{310} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*Sqrt[1 + 2*x])/155 - ((5 - 4*x)*(1 + 2*x)^(3/2))/(31*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(32678 + 10325*Sqrt[35]
))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/155 + (Sqrt[(2*(32678 +
10325*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/155 + (S
qrt[(-32678 + 10325*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/155 -
(Sqrt[(-32678 + 10325*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/155

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)^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \frac {(19-4 x) \sqrt {1+2 x}}{2+3 x+5 x^2} \, dx\\ &=-\frac {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{155} \int \frac {111+194 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {2}{155} \operatorname {Subst}\left (\int \frac {28+194 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {28 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (28-194 \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 )}{155 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {28 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (28-194 \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 )}{155 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \left (97+2 \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}{775} \left (97+2 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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 {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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} \left (2 \left (97+2 \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}{775} \left (2 \left (97+2 \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 {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \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}{155} \sqrt {\frac {2}{155} \left (32678+10325 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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 = 0.67, size = 141, normalized size = 0.50 \[ \frac {-\frac {155 \sqrt {2 x+1} (54 x+41)}{5 x^2+3 x+2}+2 \sqrt {10-5 i \sqrt {31}} \left (62-101 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+2 \sqrt {10+5 i \sqrt {31}} \left (62+101 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )}{24025} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-155*Sqrt[1 + 2*x]*(41 + 54*x))/(2 + 3*x + 5*x^2) + 2*Sqrt[10 - (5*I)*Sqrt[31]]*(62 - (101*I)*Sqrt[31])*ArcT
anh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*Sqrt[10 + (5*I)*Sqrt[31]]*(62 + (101*I)*Sqrt[31])*ArcTanh[Sqrt[5
+ 10*x]/Sqrt[2 + I*Sqrt[31]]])/24025

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fricas [B]  time = 1.03, size = 572, normalized size = 2.02 \[ \frac {1149356 \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {155} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {32678 \, \sqrt {35} + 361375} \arctan \left (\frac {1}{32833385198242899725} \cdot 5969915^{\frac {3}{4}} \sqrt {826} \sqrt {299} \sqrt {155} \sqrt {59} \sqrt {5969915^{\frac {1}{4}} \sqrt {826} \sqrt {155} {\left (97 \, \sqrt {35} \sqrt {31} - 70 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} + 41534852450 \, x + 4153485245 \, \sqrt {35} + 20767426225} \sqrt {32678 \, \sqrt {35} + 361375} {\left (2 \, \sqrt {35} - 97\right )} - \frac {1}{1715389406185} \cdot 5969915^{\frac {3}{4}} \sqrt {826} \sqrt {155} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} {\left (2 \, \sqrt {35} - 97\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 1149356 \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {155} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {32678 \, \sqrt {35} + 361375} \arctan \left (\frac {1}{1641669259912144986250} \cdot 5969915^{\frac {3}{4}} \sqrt {826} \sqrt {299} \sqrt {155} \sqrt {-147500 \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {155} {\left (97 \, \sqrt {35} \sqrt {31} - 70 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} + 6126390736375000 \, x + 612639073637500 \, \sqrt {35} + 3063195368187500} \sqrt {32678 \, \sqrt {35} + 361375} {\left (2 \, \sqrt {35} - 97\right )} - \frac {1}{1715389406185} \cdot 5969915^{\frac {3}{4}} \sqrt {826} \sqrt {155} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} {\left (2 \, \sqrt {35} - 97\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {155} {\left (32678 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} - 361375 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {32678 \, \sqrt {35} + 361375} \log \left (\frac {147500}{299} \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {155} {\left (97 \, \sqrt {35} \sqrt {31} - 70 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} + 20489601125000 \, x + 2048960112500 \, \sqrt {35} + 10244800562500\right ) - 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {155} {\left (32678 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} - 361375 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {32678 \, \sqrt {35} + 361375} \log \left (-\frac {147500}{299} \cdot 5969915^{\frac {1}{4}} \sqrt {826} \sqrt {155} {\left (97 \, \sqrt {35} \sqrt {31} - 70 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {32678 \, \sqrt {35} + 361375} + 20489601125000 \, x + 2048960112500 \, \sqrt {35} + 10244800562500\right ) - 1287580425950 \, {\left (54 \, x + 41\right )} \sqrt {2 \, x + 1}}{199574966022250 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/199574966022250*(1149356*5969915^(1/4)*sqrt(826)*sqrt(155)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(32678*sqrt(35) +
361375)*arctan(1/32833385198242899725*5969915^(3/4)*sqrt(826)*sqrt(299)*sqrt(155)*sqrt(59)*sqrt(5969915^(1/4)*
sqrt(826)*sqrt(155)*(97*sqrt(35)*sqrt(31) - 70*sqrt(31))*sqrt(2*x + 1)*sqrt(32678*sqrt(35) + 361375) + 4153485
2450*x + 4153485245*sqrt(35) + 20767426225)*sqrt(32678*sqrt(35) + 361375)*(2*sqrt(35) - 97) - 1/1715389406185*
5969915^(3/4)*sqrt(826)*sqrt(155)*sqrt(2*x + 1)*sqrt(32678*sqrt(35) + 361375)*(2*sqrt(35) - 97) + 1/31*sqrt(35
)*sqrt(31) + 2/31*sqrt(31)) + 1149356*5969915^(1/4)*sqrt(826)*sqrt(155)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(32678*
sqrt(35) + 361375)*arctan(1/1641669259912144986250*5969915^(3/4)*sqrt(826)*sqrt(299)*sqrt(155)*sqrt(-147500*59
69915^(1/4)*sqrt(826)*sqrt(155)*(97*sqrt(35)*sqrt(31) - 70*sqrt(31))*sqrt(2*x + 1)*sqrt(32678*sqrt(35) + 36137
5) + 6126390736375000*x + 612639073637500*sqrt(35) + 3063195368187500)*sqrt(32678*sqrt(35) + 361375)*(2*sqrt(3
5) - 97) - 1/1715389406185*5969915^(3/4)*sqrt(826)*sqrt(155)*sqrt(2*x + 1)*sqrt(32678*sqrt(35) + 361375)*(2*sq
rt(35) - 97) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 5969915^(1/4)*sqrt(826)*sqrt(155)*(32678*sqrt(35)*sqr
t(31)*(5*x^2 + 3*x + 2) - 361375*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(32678*sqrt(35) + 361375)*log(147500/299*5969
915^(1/4)*sqrt(826)*sqrt(155)*(97*sqrt(35)*sqrt(31) - 70*sqrt(31))*sqrt(2*x + 1)*sqrt(32678*sqrt(35) + 361375)
+ 20489601125000*x + 2048960112500*sqrt(35) + 10244800562500) - 5969915^(1/4)*sqrt(826)*sqrt(155)*(32678*sqrt
(35)*sqrt(31)*(5*x^2 + 3*x + 2) - 361375*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(32678*sqrt(35) + 361375)*log(-147500
/299*5969915^(1/4)*sqrt(826)*sqrt(155)*(97*sqrt(35)*sqrt(31) - 70*sqrt(31))*sqrt(2*x + 1)*sqrt(32678*sqrt(35)
+ 361375) + 20489601125000*x + 2048960112500*sqrt(35) + 10244800562500) - 1287580425950*(54*x + 41)*sqrt(2*x +
1))/(5*x^2 + 3*x + 2)

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giac [B]  time = 1.34, size = 624, normalized size = 2.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5768402500*sqrt(31)*(20370*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 97*sqrt(31)*(
7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 194*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 40740*(7/5)^(3/4)*sqrt
(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 68600*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 137200*(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)) + 1/5768402500*sqrt(31)*(20370*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140
*sqrt(35) + 2450) - 97*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 194*(7/5)^(3/4)*(140*sqrt(35) + 245
0)^(3/2) + 40740*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 68600*sqrt(31)*(7/5)^(1/4)*sqrt(-14
0*sqrt(35) + 2450) + 137200*(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)) + 1/11536805000*sqrt(31)*(97*sqrt(31)*(7/5)^(
3/4)*(140*sqrt(35) + 2450)^(3/2) + 20370*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 40
740*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 194*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) +
68600*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 137200*(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) - 1/11536805000*sqrt(31)*(97*sqrt(31)
*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 20370*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) -
35) - 40740*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 194*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^
(3/2) + 68600*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 137200*(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) - 4/155*(27*(2*x + 1)^(3/2)
+ 14*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)

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maple [B]  time = 0.16, size = 642, normalized size = 2.27 \[ -\frac {264 \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 )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {101 \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 )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {8 \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 )}{155 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {264 \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 )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {101 \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 )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {8 \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 )}{155 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {132 \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 )}{24025}+\frac {101 \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 )}{9610}+\frac {132 \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 )}{24025}-\frac {101 \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 )}{9610}+\frac {-\frac {108 \left (2 x +1\right )^{\frac {3}{2}}}{775}-\frac {56 \sqrt {2 x +1}}{775}}{-\frac {8 x}{5}+\left (2 x +1\right )^{2}+\frac {3}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*(-27/3100*(2*x+1)^(3/2)-7/1550*(2*x+1)^(1/2))/(-8/5*x+(2*x+1)^2+3/5)+132/24025*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)-101/9610*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)-264/4805/(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))+101/4805/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)*7^(1/2)*ar
ctan((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))+8/155/(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))-132/24025*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)+101/9610*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)-264/4805/(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))+101/4805/(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))+8/155/(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 \[ \int \frac {{\left (2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.16, size = 208, normalized size = 0.73 \[ -\frac {\frac {56\,\sqrt {2\,x+1}}{775}+\frac {108\,{\left (2\,x+1\right )}^{3/2}}{775}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}\,38272{}\mathrm {i}}{375390625\,\left (-\frac {27058304}{75078125}+\frac {\sqrt {31}\,535808{}\mathrm {i}}{75078125}\right )}-\frac {76544\,\sqrt {31}\,\sqrt {155}\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}}{11637109375\,\left (-\frac {27058304}{75078125}+\frac {\sqrt {31}\,535808{}\mathrm {i}}{75078125}\right )}\right )\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,2{}\mathrm {i}}{24025}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}\,38272{}\mathrm {i}}{375390625\,\left (\frac {27058304}{75078125}+\frac {\sqrt {31}\,535808{}\mathrm {i}}{75078125}\right )}+\frac {76544\,\sqrt {31}\,\sqrt {155}\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}}{11637109375\,\left (\frac {27058304}{75078125}+\frac {\sqrt {31}\,535808{}\mathrm {i}}{75078125}\right )}\right )\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,2{}\mathrm {i}}{24025} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(155^(1/2)*atan((155^(1/2)*(31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2)*38272i)/(375390625*((31^(1/2)*535808
i)/75078125 + 27058304/75078125)) + (76544*31^(1/2)*155^(1/2)*(31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2))/
(11637109375*((31^(1/2)*535808i)/75078125 + 27058304/75078125)))*(31^(1/2)*9269i - 32678)^(1/2)*2i)/24025 - (1
55^(1/2)*atan((155^(1/2)*(- 31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2)*38272i)/(375390625*((31^(1/2)*535808
i)/75078125 - 27058304/75078125)) - (76544*31^(1/2)*155^(1/2)*(- 31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2)
)/(11637109375*((31^(1/2)*535808i)/75078125 - 27058304/75078125)))*(- 31^(1/2)*9269i - 32678)^(1/2)*2i)/24025
- ((56*(2*x + 1)^(1/2))/775 + (108*(2*x + 1)^(3/2))/775)/((2*x + 1)^2 - (8*x)/5 + 3/5)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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