3.2328 \(\int \frac {1}{\sqrt {1+2 x} (2+3 x+5 x^2)^3} \, dx\)
Optimal. Leaf size=314 \[ \frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (7920 x+9227)}{94178 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{94178}-\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{47089} \]
[Out]
1/434*(37+20*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)^2+1/94178*(9227+7920*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)-3/20436626*arc
tan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(7379+264*35^(1/2))*(868+434*35^(1/2))
^(1/2)+3/20436626*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(7379+264*35^(1/2)
)*(868+434*35^(1/2))^(1/2)-3/40873252*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-108561594148+
28071651650*35^(1/2))^(1/2)+3/40873252*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-108561594148
+28071651650*35^(1/2))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.39, antiderivative size = 314, 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
= {740, 822, 826, 1169, 634, 618, 204, 628} \[ \frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (7920 x+9227)}{94178 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{94178}-\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{47089} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^3),x]
[Out]
(Sqrt[1 + 2*x]*(37 + 20*x))/(434*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(9227 + 7920*x))/(94178*(2 + 3*x + 5*x^
2)) - (3*Sqrt[(2 + Sqrt[35])/434]*(7379 + 264*Sqrt[35])*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sq
rt[10*(-2 + Sqrt[35])]])/47089 + (3*Sqrt[(2 + Sqrt[35])/434]*(7379 + 264*Sqrt[35])*ArcTan[(Sqrt[10*(2 + Sqrt[3
5])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/47089 - (3*Sqrt[(-250141922 + 64681225*Sqrt[35])/434]*Log[
Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/94178 + (3*Sqrt[(-250141922 + 64681225*Sqrt[3
5])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/94178
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 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}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {1}{434} \int \frac {271+100 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {26097+7920 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{94178}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {44274+7920 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{47089}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {44274 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (44274-1584 \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 )}{94178 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {44274 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (44274-1584 \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 )}{94178 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (9240+7379 \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 )}{3296230}+\frac {\left (3 \left (9240+7379 \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 )}{3296230}-\frac {\left (3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \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 )}{94178}+\frac {\left (3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \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 )}{94178}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}-\frac {\left (3 \left (9240+7379 \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 )}{1648115}-\frac {\left (3 \left (9240+7379 \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 )}{1648115}\\ &=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (250141922+64681225 \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 )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (250141922+64681225 \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 )}{47089}-\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}\\ \end {align*}
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Mathematica [C] time = 0.64, size = 151, normalized size = 0.48 \[ \frac {\frac {1085 \sqrt {2 x+1} \left (39600 x^3+69895 x^2+47861 x+26483\right )}{\left (5 x^2+3 x+2\right )^2}+6 \sqrt {10-5 i \sqrt {31}} \left (228749-23998 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+6 \sqrt {10+5 i \sqrt {31}} \left (228749+23998 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )}{102183130} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^3),x]
[Out]
((1085*Sqrt[1 + 2*x]*(26483 + 47861*x + 69895*x^2 + 39600*x^3))/(2 + 3*x + 5*x^2)^2 + 6*Sqrt[10 - (5*I)*Sqrt[3
1]]*(228749 - (23998*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 6*Sqrt[10 + (5*I)*Sqrt[31]]*(
228749 + (23998*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/102183130
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fricas [B] time = 0.79, size = 653, normalized size = 2.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
[Out]
-1/121786816718039847492020*(19347824532*97578096035^(1/4)*sqrt(105602)*sqrt(217)*sqrt(35)*(25*x^4 + 30*x^3 +
29*x^2 + 12*x + 4)*sqrt(250141922*sqrt(35) + 2263842875)*arctan(1/15121769925583791519919258475683975*97578096
035^(3/4)*sqrt(1677751)*sqrt(105602)*sqrt(37715)*sqrt(217)*sqrt(97578096035^(1/4)*sqrt(105602)*sqrt(217)*(264*
sqrt(35)*sqrt(31) - 7379*sqrt(31))*sqrt(2*x + 1)*sqrt(250141922*sqrt(35) + 2263842875) + 5959242818165770*x +
595924281816577*sqrt(35) + 2979621409082885)*sqrt(250141922*sqrt(35) + 2263842875)*(7379*sqrt(35) - 9240) - 1/
1101288930146897876195*97578096035^(3/4)*sqrt(105602)*sqrt(217)*sqrt(2*x + 1)*sqrt(250141922*sqrt(35) + 226384
2875)*(7379*sqrt(35) - 9240) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 19347824532*97578096035^(1/4)*sqrt(10
5602)*sqrt(217)*sqrt(35)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(250141922*sqrt(35) + 2263842875)*arctan(1/
389007531335643036849922924286970256875*97578096035^(3/4)*sqrt(1677751)*sqrt(105602)*sqrt(217)*sqrt(-249588676
96875*97578096035^(1/4)*sqrt(105602)*sqrt(217)*(264*sqrt(35)*sqrt(31) - 7379*sqrt(31))*sqrt(2*x + 1)*sqrt(2501
41922*sqrt(35) + 2263842875) + 148735953072151976291860968750*x + 14873595307215197629186096875*sqrt(35) + 743
67976536075988145930484375)*sqrt(250141922*sqrt(35) + 2263842875)*(7379*sqrt(35) - 9240) - 1/11012889301468978
76195*97578096035^(3/4)*sqrt(105602)*sqrt(217)*sqrt(2*x + 1)*sqrt(250141922*sqrt(35) + 2263842875)*(7379*sqrt(
35) - 9240) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) - 3*97578096035^(1/4)*sqrt(105602)*sqrt(217)*(250141922*
sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 2263842875*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x
+ 4))*sqrt(250141922*sqrt(35) + 2263842875)*log(24958867696875/1677751*97578096035^(1/4)*sqrt(105602)*sqrt(21
7)*(264*sqrt(35)*sqrt(31) - 7379*sqrt(31))*sqrt(2*x + 1)*sqrt(250141922*sqrt(35) + 2263842875) + 8865198296538
1618781250*x + 8865198296538161878125*sqrt(35) + 44325991482690809390625) + 3*97578096035^(1/4)*sqrt(105602)*s
qrt(217)*(250141922*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 2263842875*sqrt(31)*(25*x^4 + 30
*x^3 + 29*x^2 + 12*x + 4))*sqrt(250141922*sqrt(35) + 2263842875)*log(-24958867696875/1677751*97578096035^(1/4)
*sqrt(105602)*sqrt(217)*(264*sqrt(35)*sqrt(31) - 7379*sqrt(31))*sqrt(2*x + 1)*sqrt(250141922*sqrt(35) + 226384
2875) + 88651982965381618781250*x + 8865198296538161878125*sqrt(35) + 44325991482690809390625) - 1293155691541
972090*(39600*x^3 + 69895*x^2 + 47861*x + 26483)*sqrt(2*x + 1))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
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giac [B] time = 0.99, size = 642, normalized size = 2.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")
[Out]
3/175244067950*sqrt(31)*(13860*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 66*sqrt(31)
*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 132*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 27720*(7/5)^(3/4)*sq
rt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 1807855*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 3615710*
(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)) + 3/175244067950*sqrt(31)*(13860*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sq
rt(-140*sqrt(35) + 2450) - 66*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 132*(7/5)^(3/4)*(140*sqrt(35
) + 2450)^(3/2) + 27720*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 1807855*sqrt(31)*(7/5)^(1/4)
*sqrt(-140*sqrt(35) + 2450) + 3615710*(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)) + 3/350488135900*sqrt(31)*(66*sqrt(
31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 13860*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35)
- 35) - 27720*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 132*(7/5)^(3/4)*(-140*sqrt(35) + 245
0)^(3/2) + 1807855*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 3615710*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2
450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 3/350488135900*sqrt(3
1)*(66*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 13860*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)
*(2*sqrt(35) - 35) - 27720*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 132*(7/5)^(3/4)*(-140*sq
rt(35) + 2450)^(3/2) + 1807855*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 3615710*(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) + 2/47089*
(19800*(2*x + 1)^(7/2) + 10495*(2*x + 1)^(5/2) + 15332*(2*x + 1)^(3/2) + 60305*sqrt(2*x + 1))/(5*(2*x + 1)^2 -
8*x + 3)^2
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maple [B] time = 1.68, size = 1100, normalized size = 3.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(2*x+1)^(1/2)/(5*x^2+3*x+2)^3,x)
[Out]
5/20436626*(6/1353025*(-6045943503600+620096769600*5^(1/2)*7^(1/2))/(-390+40*5^(1/2)*7^(1/2))*(2*x+1)^(3/2)+1/
6765125/(-390+40*5^(1/2)*7^(1/2))*(-91048526818200*5^(1/2)+65791327714000*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)
*(2*x+1)+2/6765125*(-59423591568600*5^(1/2)*7^(1/2)+320925328420550)/(-390+40*5^(1/2)*7^(1/2))*(2*x+1)^(1/2)+1
/13530250*(-123371070933600*7^(1/2)+152992435939000*5^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)/(-390+40*5^(1/2)*7^(1
/2)))/(2*x+1/5*5^(1/2)*7^(1/2)+1/5*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+1)^2-6065475/5839036/(20*
5^(1/2)*7^(1/2)-195)*5^(1/2)*(2*35^(1/2)+4)^(1/2)*ln(10*x+5+35^(1/2)+(2*x+1)^(1/2)*(20+10*35^(1/2))^(1/2))+153
21765/20436626/(20*5^(1/2)*7^(1/2)-195)*(2*35^(1/2)+4)^(1/2)*7^(1/2)*ln(10*x+5+35^(1/2)+(2*x+1)^(1/2)*(20+10*3
5^(1/2))^(1/2))+6065475/2919518/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*(20+10*35^(1/2))^(1/2)*5^(1/2
)*(2*35^(1/2)+4)^(1/2)*arctan((10*(2*x+1)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))-15321765/1021
8313/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*(20+10*35^(1/2))^(1/2)*(2*35^(1/2)+4)^(1/2)*7^(1/2)*arct
an((10*(2*x+1)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))-8633430/329623/(20*5^(1/2)*7^(1/2)-195)/
(-20+10*35^(1/2))^(1/2)*35^(1/2)*arctan((10*(2*x+1)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))+442
7400/47089/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*arctan((10*(2*x+1)^(1/2)+(20+10*35^(1/2))^(1/2))/(
-20+10*35^(1/2))^(1/2))-5/20436626*(-6/1353025*(-6045943503600+620096769600*5^(1/2)*7^(1/2))/(-390+40*5^(1/2)*
7^(1/2))*(2*x+1)^(3/2)+1/6765125/(-390+40*5^(1/2)*7^(1/2))*(-91048526818200*5^(1/2)+65791327714000*7^(1/2))*(2
*5^(1/2)*7^(1/2)+4)^(1/2)*(2*x+1)-2/6765125*(-59423591568600*5^(1/2)*7^(1/2)+320925328420550)/(-390+40*5^(1/2)
*7^(1/2))*(2*x+1)^(1/2)+1/13530250*(-123371070933600*7^(1/2)+152992435939000*5^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1
/2)/(-390+40*5^(1/2)*7^(1/2)))/(2*x+1/5*5^(1/2)*7^(1/2)-1/5*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+
1)^2+6065475/5839036/(20*5^(1/2)*7^(1/2)-195)*5^(1/2)*(2*35^(1/2)+4)^(1/2)*ln(10*x+5+35^(1/2)-(2*x+1)^(1/2)*(2
0+10*35^(1/2))^(1/2))-15321765/20436626/(20*5^(1/2)*7^(1/2)-195)*(2*35^(1/2)+4)^(1/2)*7^(1/2)*ln(10*x+5+35^(1/
2)-(2*x+1)^(1/2)*(20+10*35^(1/2))^(1/2))+6065475/2919518/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*(20+
10*35^(1/2))^(1/2)*5^(1/2)*(2*35^(1/2)+4)^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+10*(2*x+1)^(1/2))/(-20+10*35^(
1/2))^(1/2))-15321765/10218313/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*(20+10*35^(1/2))^(1/2)*(2*35^(
1/2)+4)^(1/2)*7^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+10*(2*x+1)^(1/2))/(-20+10*35^(1/2))^(1/2))-8633430/32962
3/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*35^(1/2)*arctan((-(20+10*35^(1/2))^(1/2)+10*(2*x+1)^(1/2))/
(-20+10*35^(1/2))^(1/2))+4427400/47089/(20*5^(1/2)*7^(1/2)-195)/(-20+10*35^(1/2))^(1/2)*arctan((-(20+10*35^(1/
2))^(1/2)+10*(2*x+1)^(1/2))/(-20+10*35^(1/2))^(1/2))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} \sqrt {2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x + 1)), x)
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mupad [B] time = 0.15, size = 246, normalized size = 0.78 \[ -\frac {\frac {3446\,\sqrt {2\,x+1}}{33635}+\frac {30664\,{\left (2\,x+1\right )}^{3/2}}{1177225}+\frac {4198\,{\left (2\,x+1\right )}^{5/2}}{235445}+\frac {1584\,{\left (2\,x+1\right )}^{7/2}}{47089}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{665489348040125\,\left (\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}+\frac {46760544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{20630169789243875\,\left (\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}\right )\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{10218313}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{665489348040125\,\left (-\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}-\frac {46760544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{20630169789243875\,\left (-\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}\right )\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{10218313} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((2*x + 1)^(1/2)*(3*x + 5*x^2 + 2)^3),x)
[Out]
(217^(1/2)*atan((217^(1/2)*(- 31^(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*23380272i)/(66548934804012
5*((31^(1/2)*172523027088i)/95069906862875 + 561079767456/95069906862875)) + (46760544*31^(1/2)*217^(1/2)*(- 3
1^(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2))/(20630169789243875*((31^(1/2)*172523027088i)/95069906862
875 + 561079767456/95069906862875)))*(- 31^(1/2)*52010281i - 250141922)^(1/2)*3i)/10218313 - ((3446*(2*x + 1)^
(1/2))/33635 + (30664*(2*x + 1)^(3/2))/1177225 + (4198*(2*x + 1)^(5/2))/235445 + (1584*(2*x + 1)^(7/2))/47089)
/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) - (217^(1/2)*atan((217^(1/2)*(31^
(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*23380272i)/(665489348040125*((31^(1/2)*172523027088i)/95069
906862875 - 561079767456/95069906862875)) - (46760544*31^(1/2)*217^(1/2)*(31^(1/2)*52010281i - 250141922)^(1/2
)*(2*x + 1)^(1/2))/(20630169789243875*((31^(1/2)*172523027088i)/95069906862875 - 561079767456/95069906862875))
)*(31^(1/2)*52010281i - 250141922)^(1/2)*3i)/10218313
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 x + 1} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+2*x)**(1/2)/(5*x**2+3*x+2)**3,x)
[Out]
Integral(1/(sqrt(2*x + 1)*(5*x**2 + 3*x + 2)**3), x)
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