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

Optimal. Leaf size=313 \[ \frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}+\frac {5 (2080 x+2329)}{94178 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}-\frac {81090}{329623 \sqrt {2 x+1}}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{659246}-\frac {15 \sqrt {\frac {1}{434} \left (387427075 \sqrt {35}-2257111762\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (387427075 \sqrt {35}-2257111762\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{329623} \]

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Rubi [A]  time = 0.42, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {740, 822, 828, 826, 1169, 634, 618, 204, 628} \begin {gather*} \frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}+\frac {5 (2080 x+2329)}{94178 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}-\frac {81090}{329623 \sqrt {2 x+1}}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{659246}-\frac {15 \sqrt {\frac {1}{434} \left (387427075 \sqrt {35}-2257111762\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 )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (387427075 \sqrt {35}-2257111762\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 )}{329623} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-81090/(329623*Sqrt[1 + 2*x]) + (37 + 20*x)/(434*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^2) + (5*(2329 + 2080*x))/(941
78*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) - (15*Sqrt[(-2257111762 + 387427075*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sq
rt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/329623 + (15*Sqrt[(-2257111762 + 387427075*Sqrt[35])/4
34]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/329623 - (15*Sqrt[(22571117
62 + 387427075*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/659246 + (1
5*Sqrt[(2257111762 + 387427075*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*
x)])/659246

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 740

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

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (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 828

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

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}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {1}{434} \int \frac {345+140 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {56145+31200 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{94178}\\ &=-\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {68655-405450 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{659246}\\ &=-\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {542760-405450 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{329623}\\ &=-\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {108552 \sqrt {10 \left (2+\sqrt {35}\right )}-\left (542760+81090 \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 )}{659246 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {108552 \sqrt {10 \left (2+\sqrt {35}\right )}+\left (542760+81090 \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 )}{659246 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {\left (3 \left (94605-18092 \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 )}{4614722}-\frac {\left (3 \left (94605-18092 \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 )}{4614722}-\frac {\left (15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \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 )}{659246}+\frac {\left (15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \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 )}{659246}\\ &=-\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {\left (3 \left (94605-18092 \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 )}{2307361}+\frac {\left (3 \left (94605-18092 \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 )}{2307361}\\ &=-\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \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 )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \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 )}{329623}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}\\ \end {align*}

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Mathematica [C]  time = 0.86, size = 156, normalized size = 0.50 \begin {gather*} \frac {-\frac {217 \left (4054500 x^4+4501400 x^3+4077245 x^2+1525635 x+429487\right )}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}+6 \sqrt {10-5 i \sqrt {31}} \left (560852+58421 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+6 \sqrt {10+5 i \sqrt {31}} \left (560852-58421 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )}{143056382} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-217*(429487 + 1525635*x + 4077245*x^2 + 4501400*x^3 + 4054500*x^4))/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^2) + 6
*Sqrt[10 - (5*I)*Sqrt[31]]*(560852 + (58421*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 6*Sqrt
[10 + (5*I)*Sqrt[31]]*(560852 - (58421*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/143056382

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IntegrateAlgebraic [C]  time = 2.62, size = 176, normalized size = 0.56 \begin {gather*} -\frac {2 \left (1013625 (2 x+1)^4-1803800 (2 x+1)^3+3406895 (2 x+1)^2-2405620 (2 x+1)+1506848\right )}{329623 \sqrt {2 x+1} \left (5 (2 x+1)^2-4 (2 x+1)+7\right )^2}+\frac {15 \sqrt {\frac {1}{217} \left (-2257111762+71603149 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}-\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )}{329623}+\frac {15 \sqrt {\frac {1}{217} \left (-2257111762-71603149 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}+\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )}{329623} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1506848 - 2405620*(1 + 2*x) + 3406895*(1 + 2*x)^2 - 1803800*(1 + 2*x)^3 + 1013625*(1 + 2*x)^4))/(329623*S
qrt[1 + 2*x]*(7 - 4*(1 + 2*x) + 5*(1 + 2*x)^2)^2) + (15*Sqrt[(-2257111762 + (71603149*I)*Sqrt[31])/217]*ArcTan
[Sqrt[-2/7 - (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/329623 + (15*Sqrt[(-2257111762 - (71603149*I)*Sqrt[31])/217]*ArcT
an[Sqrt[-2/7 + (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]])/329623

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fricas [B]  time = 0.46, size = 657, normalized size = 2.10

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/49209733431234418124460140*(26636371428*3500868535115^(1/4)*sqrt(217)*sqrt(35)*(50*x^5 + 85*x^4 + 88*x^3 +
53*x^2 + 20*x + 4)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750)*arctan(1/15090017683016917385677
87671747064606625*3500868535115^(3/4)*sqrt(2309779)*sqrt(316267)*sqrt(217)*sqrt(3500868535115^(1/4)*sqrt(217)*
(2703*sqrt(35)*sqrt(31) + 18092*sqrt(31))*sqrt(2*x + 1)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893
750) + 1719941911827268850*x + 171994191182726885*sqrt(35) + 859970955913634425)*(18092*sqrt(35) + 94605)*sqrt
(-1748932415799512300*sqrt(35) + 10506981691013893750) - 1/1903863040197561930840325*3500868535115^(3/4)*sqrt(
217)*sqrt(2*x + 1)*(18092*sqrt(35) + 94605)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750) - 1/31*
sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 26636371428*3500868535115^(1/4)*sqrt(217)*sqrt(35)*(50*x^5 + 85*x^4 + 88*
x^3 + 53*x^2 + 20*x + 4)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750)*arctan(1/27727907492543585
696183098468352312146734375*3500868535115^(3/4)*sqrt(2309779)*sqrt(217)*sqrt(-106784587546875*3500868535115^(1
/4)*sqrt(217)*(2703*sqrt(35)*sqrt(31) + 18092*sqrt(31))*sqrt(2*x + 1)*sqrt(-1748932415799512300*sqrt(35) + 105
06981691013893750) + 183663287659058552515752602343750*x + 18366328765905855251575260234375*sqrt(35) + 9183164
3829529276257876301171875)*(18092*sqrt(35) + 94605)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750)
- 1/1903863040197561930840325*3500868535115^(3/4)*sqrt(217)*sqrt(2*x + 1)*(18092*sqrt(35) + 94605)*sqrt(-1748
932415799512300*sqrt(35) + 10506981691013893750) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) - 3*3500868535115^(
1/4)*sqrt(217)*(2257111762*sqrt(35)*sqrt(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4) + 13559947625*sqrt
(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4))*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750
)*log(106784587546875/2309779*3500868535115^(1/4)*sqrt(217)*(2703*sqrt(35)*sqrt(31) + 18092*sqrt(31))*sqrt(2*x
+ 1)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750) + 79515524064881771163281250*x + 795155240648
8177116328125*sqrt(35) + 39757762032440885581640625) + 3*3500868535115^(1/4)*sqrt(217)*(2257111762*sqrt(35)*sq
rt(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4) + 13559947625*sqrt(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^
2 + 20*x + 4))*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750)*log(-106784587546875/2309779*3500868
535115^(1/4)*sqrt(217)*(2703*sqrt(35)*sqrt(31) + 18092*sqrt(31))*sqrt(2*x + 1)*sqrt(-1748932415799512300*sqrt(
35) + 10506981691013893750) + 79515524064881771163281250*x + 7951552406488177116328125*sqrt(35) + 397577620324
40885581640625) + 74645478973303468090*(4054500*x^4 + 4501400*x^3 + 4077245*x^2 + 1525635*x + 429487)*sqrt(2*x
+ 1))/(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)

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giac [B]  time = 1.05, size = 651, normalized size = 2.08

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/981366780520*sqrt(31)*(567630*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 2703*sqrt
(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 5406*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 1135260*(7/5)^(
3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 17730160*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) -
35460320*(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)) - 3/981366780520*sqrt(31)*(567630*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35
) + 35)*sqrt(-140*sqrt(35) + 2450) - 2703*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 5406*(7/5)^(3/4)
*(140*sqrt(35) + 2450)^(3/2) + 1135260*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 17730160*sqrt
(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 35460320*(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)) - 3/1962733561040*
sqrt(31)*(2703*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 567630*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35
) + 2450)*(2*sqrt(35) - 35) - 1135260*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 5406*(7/5)^(3
/4)*(-140*sqrt(35) + 2450)^(3/2) - 17730160*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 35460320*(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) + 3/1962733561040*sqrt(31)*(2703*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 567630*sqrt(31)*(7/5)^(
3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 1135260*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) +
2450) + 5406*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 17730160*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450
) + 35460320*(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) - 64/343/sqrt(2*x + 1) - 2/47089*(34975*(2*x + 1)^(7/2) - 81960*(2*x + 1)^(5/2) + 108
889*(2*x + 1)^(3/2) - 97644*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)^2

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maple [B]  time = 0.16, size = 671, normalized size = 2.14 \begin {gather*} -\frac {475725 \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 )}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {876315 \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 )}{143056382 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {542760 \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 )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {475725 \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 )}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {876315 \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 )}{143056382 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {542760 \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 )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {95145 \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 )}{20436626}-\frac {876315 \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 )}{286112764}+\frac {95145 \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 )}{20436626}+\frac {876315 \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 )}{286112764}-\frac {1600 \left (\frac {9793 \left (2 x +1\right )^{\frac {7}{2}}}{30752}-\frac {14343 \left (2 x +1\right )^{\frac {5}{2}}}{19220}+\frac {762223 \left (2 x +1\right )^{\frac {3}{2}}}{768800}-\frac {170877 \sqrt {2 x +1}}{192200}\right )}{343 \left (-8 x +5 \left (2 x +1\right )^{2}+3\right )^{2}}-\frac {64}{343 \sqrt {2 x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1600/343*(9793/30752*(2*x+1)^(7/2)-14343/19220*(2*x+1)^(5/2)+762223/768800*(2*x+1)^(3/2)-170877/192200*(2*x+1
)^(1/2))/(-8*x+5*(2*x+1)^2+3)^2+95145/20436626*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)+876315/286112764*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)-475725/10218313/(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))-876315/143056382/(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))+542760/2307361/(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))-95145/20436626*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)-876315/286112764*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)-475725/10218313/(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))-876315/
143056382/(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))+542760/2307361/(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))-64/
343/(2*x+1)^(1/2)

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

[Out]

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

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mupad [B]  time = 0.18, size = 252, normalized size = 0.81 \begin {gather*} \frac {\frac {274928\,x}{235445}-\frac {1362758\,{\left (2\,x+1\right )}^2}{1648115}+\frac {144304\,{\left (2\,x+1\right )}^3}{329623}-\frac {81090\,{\left (2\,x+1\right )}^4}{329623}+\frac {256792}{1177225}}{\frac {49\,\sqrt {2\,x+1}}{25}-\frac {56\,{\left (2\,x+1\right )}^{3/2}}{25}+\frac {86\,{\left (2\,x+1\right )}^{5/2}}{25}-\frac {8\,{\left (2\,x+1\right )}^{7/2}}{5}+{\left (2\,x+1\right )}^{9/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}\,32187888{}\mathrm {i}}{1826102771022103\,\left (-\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}+\frac {64375776\,\sqrt {31}\,\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}}{56609185901685193\,\left (-\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}\right )\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}\,15{}\mathrm {i}}{71528191}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}\,32187888{}\mathrm {i}}{1826102771022103\,\left (\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}-\frac {64375776\,\sqrt {31}\,\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}}{56609185901685193\,\left (\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}\right )\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}\,15{}\mathrm {i}}{71528191} \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)^3),x)

[Out]

((274928*x)/235445 - (1362758*(2*x + 1)^2)/1648115 + (144304*(2*x + 1)^3)/329623 - (81090*(2*x + 1)^4)/329623
+ 256792/1177225)/((49*(2*x + 1)^(1/2))/25 - (56*(2*x + 1)^(3/2))/25 + (86*(2*x + 1)^(5/2))/25 - (8*(2*x + 1)^
(7/2))/5 + (2*x + 1)^(9/2)) + (217^(1/2)*atan((217^(1/2)*(2*x + 1)^(1/2)*(2257111762 - 31^(1/2)*71603149i)^(1/
2)*32187888i)/(1826102771022103*((31^(1/2)*582343269696i)/260871824431729 - 1880448604848/260871824431729)) +
(64375776*31^(1/2)*217^(1/2)*(2*x + 1)^(1/2)*(2257111762 - 31^(1/2)*71603149i)^(1/2))/(56609185901685193*((31^
(1/2)*582343269696i)/260871824431729 - 1880448604848/260871824431729)))*(2257111762 - 31^(1/2)*71603149i)^(1/2
)*15i)/71528191 - (217^(1/2)*atan((217^(1/2)*(2*x + 1)^(1/2)*(31^(1/2)*71603149i + 2257111762)^(1/2)*32187888i
)/(1826102771022103*((31^(1/2)*582343269696i)/260871824431729 + 1880448604848/260871824431729)) - (64375776*31
^(1/2)*217^(1/2)*(2*x + 1)^(1/2)*(31^(1/2)*71603149i + 2257111762)^(1/2))/(56609185901685193*((31^(1/2)*582343
269696i)/260871824431729 + 1880448604848/260871824431729)))*(31^(1/2)*71603149i + 2257111762)^(1/2)*15i)/71528
191

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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}} \left (5 x^{2} + 3 x + 2\right )^{3}}\, 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)**3,x)

[Out]

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

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