3.24.14 \(\int \frac {1}{(1+2 x)^{3/2} (2+3 x+5 x^2)} \, dx\) [2314]

Optimal. Leaf size=253 \[ -\frac {4}{7 \sqrt {1+2 x}}+\frac {1}{7} \sqrt {\frac {2}{217} \left (-178+35 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{7} \sqrt {\frac {2}{217} \left (-178+35 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{7} \sqrt {\frac {1}{434} \left (178+35 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{7} \sqrt {\frac {1}{434} \left (178+35 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

-4/7/(1+2*x)^(1/2)+1/1519*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-77252+1
5190*35^(1/2))^(1/2)-1/1519*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-77252+
15190*35^(1/2))^(1/2)-1/3038*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(77252+15190*35^(1/2))^(
1/2)+1/3038*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(77252+15190*35^(1/2))^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {723, 840, 1183, 648, 632, 210, 642} \begin {gather*} \frac {1}{7} \sqrt {\frac {2}{217} \left (35 \sqrt {35}-178\right )} \text {ArcTan}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{7} \sqrt {\frac {2}{217} \left (35 \sqrt {35}-178\right )} \text {ArcTan}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {4}{7 \sqrt {2 x+1}}-\frac {1}{7} \sqrt {\frac {1}{434} \left (178+35 \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}{7} \sqrt {\frac {1}{434} \left (178+35 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-4/(7*Sqrt[1 + 2*x]) + (Sqrt[(2*(-178 + 35*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])
/Sqrt[10*(-2 + Sqrt[35])]])/7 - (Sqrt[(2*(-178 + 35*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[
1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/7 - (Sqrt[(178 + 35*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]
*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/7 + (Sqrt[(178 + 35*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/7

Rule 210

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

Rule 632

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 642

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 648

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 723

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

Rule 840

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 1183

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}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx &=-\frac {4}{7 \sqrt {1+2 x}}+\frac {1}{7} \int \frac {-1-10 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {4}{7 \sqrt {1+2 x}}+\frac {2}{7} \text {Subst}\left (\int \frac {8-10 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\frac {4}{7 \sqrt {1+2 x}}+\frac {\text {Subst}\left (\int \frac {8 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (8+2 \sqrt {35}\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 )}{7 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {8 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (8+2 \sqrt {35}\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 )}{7 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {4}{7 \sqrt {1+2 x}}-\frac {\left (4+\sqrt {35}\right ) \text {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 )}{7 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\left (4+\sqrt {35}\right ) \text {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 )}{7 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {1}{245} \left (-35+4 \sqrt {35}\right ) \text {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}{245} \left (-35+4 \sqrt {35}\right ) \text {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 {4}{7 \sqrt {1+2 x}}-\frac {1}{7} \sqrt {\frac {89}{217}+\frac {5 \sqrt {35}}{62}} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{7} \sqrt {\frac {89}{217}+\frac {5 \sqrt {35}}{62}} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{245} \left (2 \left (35-4 \sqrt {35}\right )\right ) \text {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}{245} \left (2 \left (35-4 \sqrt {35}\right )\right ) \text {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 {4}{7 \sqrt {1+2 x}}+\frac {1}{7} \sqrt {\frac {2}{217} \left (-178+35 \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}{7} \sqrt {\frac {2}{217} \left (-178+35 \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}{7} \sqrt {\frac {89}{217}+\frac {5 \sqrt {35}}{62}} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{7} \sqrt {\frac {89}{217}+\frac {5 \sqrt {35}}{62}} \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] Result contains complex when optimal does not.
time = 0.52, size = 114, normalized size = 0.45 \begin {gather*} \frac {2 \left (-\frac {434}{\sqrt {1+2 x}}-\sqrt {217 \left (-178-19 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-\sqrt {217 \left (-178+19 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{1519} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-434/Sqrt[1 + 2*x] - Sqrt[217*(-178 - (19*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] -
Sqrt[217*(-178 + (19*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/1519

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(388\) vs. \(2(167)=334\).
time = 2.38, size = 389, normalized size = 1.54 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/7/(2*x+1)^(1/2)+1/15190*(135*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+70*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))*ln
(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2))+2/1519*(248*5^(1/2)*7^(1/2)-1/10*(1
35*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+70*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(
1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(2*x+1)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2)*
7^(1/2)-20)^(1/2))+1/15190*(-135*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-70*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))*l
n(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)+2/1519*(248*5^(1/2)*7^(1/2)+1/10*
(-135*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-70*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*
5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (170) = 340\).
time = 4.69, size = 500, normalized size = 1.98 \begin {gather*} \frac {2356 \cdot 42875^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (2 \, x + 1\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \arctan \left (\frac {1}{3644188279375} \cdot 42875^{\frac {3}{4}} \sqrt {217} \sqrt {31} \sqrt {42875^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} + 4 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 1443050 \, x + 144305 \, \sqrt {35} + 721525} {\left (4 \, \sqrt {35} \sqrt {19} + 35 \, \sqrt {19}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} - \frac {1}{176773625} \cdot 42875^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} {\left (4 \, \sqrt {35} + 35\right )} \sqrt {-12460 \, \sqrt {35} + 85750} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 2356 \cdot 42875^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (2 \, x + 1\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \arctan \left (\frac {1}{255093179556250} \cdot 42875^{\frac {3}{4}} \sqrt {217} \sqrt {-151900 \cdot 42875^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} + 4 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 219199295000 \, x + 21919929500 \, \sqrt {35} + 109599647500} {\left (4 \, \sqrt {35} \sqrt {19} + 35 \, \sqrt {19}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} - \frac {1}{176773625} \cdot 42875^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} {\left (4 \, \sqrt {35} + 35\right )} \sqrt {-12460 \, \sqrt {35} + 85750} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 42875^{\frac {1}{4}} \sqrt {217} {\left (178 \, \sqrt {35} \sqrt {31} {\left (2 \, x + 1\right )} + 1225 \, \sqrt {31} {\left (2 \, x + 1\right )}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \log \left (\frac {151900}{19} \cdot 42875^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} + 4 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 11536805000 \, x + 1153680500 \, \sqrt {35} + 5768402500\right ) - 42875^{\frac {1}{4}} \sqrt {217} {\left (178 \, \sqrt {35} \sqrt {31} {\left (2 \, x + 1\right )} + 1225 \, \sqrt {31} {\left (2 \, x + 1\right )}\right )} \sqrt {-12460 \, \sqrt {35} + 85750} \log \left (-\frac {151900}{19} \cdot 42875^{\frac {1}{4}} \sqrt {217} {\left (\sqrt {35} \sqrt {31} + 4 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-12460 \, \sqrt {35} + 85750} + 11536805000 \, x + 1153680500 \, \sqrt {35} + 5768402500\right ) - 1252567400 \, \sqrt {2 \, x + 1}}{2191992950 \, {\left (2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2191992950*(2356*42875^(1/4)*sqrt(217)*sqrt(35)*(2*x + 1)*sqrt(-12460*sqrt(35) + 85750)*arctan(1/36441882793
75*42875^(3/4)*sqrt(217)*sqrt(31)*sqrt(42875^(1/4)*sqrt(217)*(sqrt(35)*sqrt(31) + 4*sqrt(31))*sqrt(2*x + 1)*sq
rt(-12460*sqrt(35) + 85750) + 1443050*x + 144305*sqrt(35) + 721525)*(4*sqrt(35)*sqrt(19) + 35*sqrt(19))*sqrt(-
12460*sqrt(35) + 85750) - 1/176773625*42875^(3/4)*sqrt(217)*sqrt(2*x + 1)*(4*sqrt(35) + 35)*sqrt(-12460*sqrt(3
5) + 85750) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 2356*42875^(1/4)*sqrt(217)*sqrt(35)*(2*x + 1)*sqrt(-12
460*sqrt(35) + 85750)*arctan(1/255093179556250*42875^(3/4)*sqrt(217)*sqrt(-151900*42875^(1/4)*sqrt(217)*(sqrt(
35)*sqrt(31) + 4*sqrt(31))*sqrt(2*x + 1)*sqrt(-12460*sqrt(35) + 85750) + 219199295000*x + 21919929500*sqrt(35)
+ 109599647500)*(4*sqrt(35)*sqrt(19) + 35*sqrt(19))*sqrt(-12460*sqrt(35) + 85750) - 1/176773625*42875^(3/4)*s
qrt(217)*sqrt(2*x + 1)*(4*sqrt(35) + 35)*sqrt(-12460*sqrt(35) + 85750) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31
)) + 42875^(1/4)*sqrt(217)*(178*sqrt(35)*sqrt(31)*(2*x + 1) + 1225*sqrt(31)*(2*x + 1))*sqrt(-12460*sqrt(35) +
85750)*log(151900/19*42875^(1/4)*sqrt(217)*(sqrt(35)*sqrt(31) + 4*sqrt(31))*sqrt(2*x + 1)*sqrt(-12460*sqrt(35)
+ 85750) + 11536805000*x + 1153680500*sqrt(35) + 5768402500) - 42875^(1/4)*sqrt(217)*(178*sqrt(35)*sqrt(31)*(
2*x + 1) + 1225*sqrt(31)*(2*x + 1))*sqrt(-12460*sqrt(35) + 85750)*log(-151900/19*42875^(1/4)*sqrt(217)*(sqrt(3
5)*sqrt(31) + 4*sqrt(31))*sqrt(2*x + 1)*sqrt(-12460*sqrt(35) + 85750) + 11536805000*x + 1153680500*sqrt(35) +
5768402500) - 1252567400*sqrt(2*x + 1))/(2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (170) = 340\).
time = 1.59, size = 594, normalized size = 2.35 \begin {gather*} -\frac {1}{52101700} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 3920 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 7840 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) - \frac {1}{52101700} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 3920 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 7840 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (-\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} - \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) - \frac {1}{104203400} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 3920 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 7840 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) + \frac {1}{104203400} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 3920 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 7840 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (-2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) - \frac {4}{7 \, \sqrt {2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/52101700*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - sqrt(31)*(7/5)^(
3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/5)^(3/4)*sqrt(140*sqrt(
35) + 2450)*(2*sqrt(35) - 35) - 3920*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 7840*(7/5)^(1/4)*sqrt(1
40*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)) - 1/52101700*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450)
- sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/5)^(
3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 3920*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 7840
*(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)) - 1/104203400*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2)
+ 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2*sqrt(35) + 35)*sq
rt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 3920*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqr
t(35) + 2450) + 7840*(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) + 1/104203400*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 210
*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-14
0*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 3920*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35)
+ 2450) + 7840*(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) - 4/7/sqrt(2*x + 1)

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Mupad [B]
time = 0.14, size = 182, normalized size = 0.72 \begin {gather*} -\frac {4}{7\,\sqrt {2\,x+1}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}\,2432{}\mathrm {i}}{2100875\,\left (\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}-\frac {4864\,\sqrt {31}\,\sqrt {217}\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}}{65127125\,\left (\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}\right )\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,2{}\mathrm {i}}{1519}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}\,2432{}\mathrm {i}}{2100875\,\left (-\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}+\frac {4864\,\sqrt {31}\,\sqrt {217}\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}}{65127125\,\left (-\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}\right )\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,2{}\mathrm {i}}{1519} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(217^(1/2)*atan((217^(1/2)*(31^(1/2)*19i + 178)^(1/2)*(2*x + 1)^(1/2)*2432i)/(2100875*((31^(1/2)*9728i)/300125
- 65664/300125)) + (4864*31^(1/2)*217^(1/2)*(31^(1/2)*19i + 178)^(1/2)*(2*x + 1)^(1/2))/(65127125*((31^(1/2)*
9728i)/300125 - 65664/300125)))*(31^(1/2)*19i + 178)^(1/2)*2i)/1519 - (217^(1/2)*atan((217^(1/2)*(178 - 31^(1/
2)*19i)^(1/2)*(2*x + 1)^(1/2)*2432i)/(2100875*((31^(1/2)*9728i)/300125 + 65664/300125)) - (4864*31^(1/2)*217^(
1/2)*(178 - 31^(1/2)*19i)^(1/2)*(2*x + 1)^(1/2))/(65127125*((31^(1/2)*9728i)/300125 + 65664/300125)))*(178 - 3
1^(1/2)*19i)^(1/2)*2i)/1519 - 4/(7*(2*x + 1)^(1/2))

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