3.24.24 \(\int \frac {(1+2 x)^{7/2}}{(2+3 x+5 x^2)^3} \, dx\) [2324]

Optimal. Leaf size=300 \[ -\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{4805}+\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{4805}-\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610}+\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610} \]

[Out]

-1/62*(5-4*x)*(1+2*x)^(5/2)/(5*x^2+3*x+2)^2-1/9610*(957-592*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)-1/2979100*ln(5+10*x
+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-2991829220+560131250*35^(1/2))^(1/2)+1/2979100*ln(5+10*x+35^
(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-2991829220+560131250*35^(1/2))^(1/2)-1/1489550*arctan((-10*(1+2*
x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(2991829220+560131250*35^(1/2))^(1/2)+1/1489550*arct
an((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(2991829220+560131250*35^(1/2))^(1/2)

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Rubi [A]
time = 0.29, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {752, 832, 840, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \sqrt {35}\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 )}{4805}+\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \sqrt {35}\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 )}{4805}-\frac {(5-4 x) (2 x+1)^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac {(957-592 x) \sqrt {2 x+1}}{9610 \left (5 x^2+3 x+2\right )}-\frac {\sqrt {\frac {1}{310} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{9610}+\frac {\sqrt {\frac {1}{310} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{9610} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

-1/62*((5 - 4*x)*(1 + 2*x)^(5/2))/(2 + 3*x + 5*x^2)^2 - ((957 - 592*x)*Sqrt[1 + 2*x])/(9610*(2 + 3*x + 5*x^2))
- (Sqrt[(9651062 + 1806875*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + S
qrt[35])]])/4805 + (Sqrt[(9651062 + 1806875*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])
/Sqrt[10*(-2 + Sqrt[35])]])/4805 - (Sqrt[(-9651062 + 1806875*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[3
5])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/9610 + (Sqrt[(-9651062 + 1806875*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 +
Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/9610

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 752

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 832

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

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+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {(1+2 x)^{3/2} (37+4 x)}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1797-1088 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{9610}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\text {Subst}\left (\int \frac {-2506-1088 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{4805}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\text {Subst}\left (\int \frac {-2506 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-2506+1088 \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 )}{9610 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\text {Subst}\left (\int \frac {-2506 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-2506+1088 \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 )}{9610 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {1417371+194752 \sqrt {35}} \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 )}{48050}+\frac {\sqrt {1417371+194752 \sqrt {35}} \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 )}{48050}-\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \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 )}{9610}+\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \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 )}{9610}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610}+\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610}-\frac {\sqrt {1417371+194752 \sqrt {35}} \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 )}{24025}-\frac {\sqrt {1417371+194752 \sqrt {35}} \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 )}{24025}\\ &=-\frac {(5-4 x) (1+2 x)^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(957-592 x) \sqrt {1+2 x}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \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 )}{4805}+\frac {\sqrt {\frac {1}{310} \left (9651062+1806875 \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 )}{4805}-\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610}+\frac {\sqrt {\frac {1}{310} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{9610}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.58, size = 141, normalized size = 0.47 \begin {gather*} \frac {\frac {155 \sqrt {1+2 x} \left (-2689-4167 x-3629 x^2+5440 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+\sqrt {155 \left (9651062+825499 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {155 \left (9651062-825499 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{744775} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

((155*Sqrt[1 + 2*x]*(-2689 - 4167*x - 3629*x^2 + 5440*x^3))/(2*(2 + 3*x + 5*x^2)^2) + Sqrt[155*(9651062 + (825
499*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + Sqrt[155*(9651062 - (825499*I)*Sqrt[31])]*
ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/744775

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(434\) vs. \(2(210)=420\).
time = 1.84, size = 435, normalized size = 1.45 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+1)^(7/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)

[Out]

1600*(17/24025*(2*x+1)^(7/2)-11789/3844000*(2*x+1)^(5/2)+1771/961000*(2*x+1)^(3/2)-8771/3844000*(2*x+1)^(1/2))
/(5*(2*x+1)^2-8*x+3)^2+1/2979100*(-7353*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+4510*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)
^(1/2))*ln(-(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)+1/148955*(11098*5^(1/2)*
7^(1/2)+1/10*(-7353*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+4510*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(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))+1/2979100*(7353*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-4510*7^(1/2)*(2*5^(
1/2)*7^(1/2)+4)^(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))+1/148955*(
11098*5^(1/2)*7^(1/2)-1/10*(7353*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-4510*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(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))

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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+2*x)^(7/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^3, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (213) = 426\).
time = 3.81, size = 652, normalized size = 2.17 \begin {gather*} -\frac {102361876 \cdot 121835^{\frac {1}{4}} \sqrt {155} \sqrt {118} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {9651062 \, \sqrt {35} + 63240625} \arctan \left (\frac {1}{5314799928145246742525} \cdot 121835^{\frac {3}{4}} \sqrt {26629} \sqrt {413} \sqrt {155} \sqrt {118} \sqrt {121835^{\frac {1}{4}} \sqrt {155} \sqrt {118} {\left (544 \, \sqrt {35} \sqrt {31} - 6265 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} + 528443184850 \, x + 52844318485 \, \sqrt {35} + 264221592425} \sqrt {9651062 \, \sqrt {35} + 63240625} {\left (179 \, \sqrt {35} - 544\right )} - \frac {1}{3117814790615} \cdot 121835^{\frac {3}{4}} \sqrt {155} \sqrt {118} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} {\left (179 \, \sqrt {35} - 544\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 102361876 \cdot 121835^{\frac {1}{4}} \sqrt {155} \sqrt {118} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {9651062 \, \sqrt {35} + 63240625} \arctan \left (\frac {1}{23252249685635454498546875} \cdot 121835^{\frac {3}{4}} \sqrt {26629} \sqrt {155} \sqrt {118} \sqrt {-7905078125 \cdot 121835^{\frac {1}{4}} \sqrt {155} \sqrt {118} {\left (544 \, \sqrt {35} \sqrt {31} - 6265 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} + 4177384660863066406250 \, x + 417738466086306640625 \, \sqrt {35} + 2088692330431533203125} \sqrt {9651062 \, \sqrt {35} + 63240625} {\left (179 \, \sqrt {35} - 544\right )} - \frac {1}{3117814790615} \cdot 121835^{\frac {3}{4}} \sqrt {155} \sqrt {118} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} {\left (179 \, \sqrt {35} - 544\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) - 121835^{\frac {1}{4}} \sqrt {155} \sqrt {118} {\left (9651062 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 63240625 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {9651062 \, \sqrt {35} + 63240625} \log \left (\frac {7905078125}{26629} \cdot 121835^{\frac {1}{4}} \sqrt {155} \sqrt {118} {\left (544 \, \sqrt {35} \sqrt {31} - 6265 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} + 156873508613281250 \, x + 15687350861328125 \, \sqrt {35} + 78436754306640625\right ) + 121835^{\frac {1}{4}} \sqrt {155} \sqrt {118} {\left (9651062 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 63240625 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {9651062 \, \sqrt {35} + 63240625} \log \left (-\frac {7905078125}{26629} \cdot 121835^{\frac {1}{4}} \sqrt {155} \sqrt {118} {\left (544 \, \sqrt {35} \sqrt {31} - 6265 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {9651062 \, \sqrt {35} + 63240625} + 156873508613281250 \, x + 15687350861328125 \, \sqrt {35} + 78436754306640625\right ) - 16381738730350 \, {\left (5440 \, x^{3} - 3629 \, x^{2} - 4167 \, x - 2689\right )} \sqrt {2 \, x + 1}}{157428509198663500 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")

[Out]

-1/157428509198663500*(102361876*121835^(1/4)*sqrt(155)*sqrt(118)*sqrt(35)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x +
4)*sqrt(9651062*sqrt(35) + 63240625)*arctan(1/5314799928145246742525*121835^(3/4)*sqrt(26629)*sqrt(413)*sqrt(1
55)*sqrt(118)*sqrt(121835^(1/4)*sqrt(155)*sqrt(118)*(544*sqrt(35)*sqrt(31) - 6265*sqrt(31))*sqrt(2*x + 1)*sqrt
(9651062*sqrt(35) + 63240625) + 528443184850*x + 52844318485*sqrt(35) + 264221592425)*sqrt(9651062*sqrt(35) +
63240625)*(179*sqrt(35) - 544) - 1/3117814790615*121835^(3/4)*sqrt(155)*sqrt(118)*sqrt(2*x + 1)*sqrt(9651062*s
qrt(35) + 63240625)*(179*sqrt(35) - 544) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 102361876*121835^(1/4)*sq
rt(155)*sqrt(118)*sqrt(35)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(9651062*sqrt(35) + 63240625)*arctan(1/23
252249685635454498546875*121835^(3/4)*sqrt(26629)*sqrt(155)*sqrt(118)*sqrt(-7905078125*121835^(1/4)*sqrt(155)*
sqrt(118)*(544*sqrt(35)*sqrt(31) - 6265*sqrt(31))*sqrt(2*x + 1)*sqrt(9651062*sqrt(35) + 63240625) + 4177384660
863066406250*x + 417738466086306640625*sqrt(35) + 2088692330431533203125)*sqrt(9651062*sqrt(35) + 63240625)*(1
79*sqrt(35) - 544) - 1/3117814790615*121835^(3/4)*sqrt(155)*sqrt(118)*sqrt(2*x + 1)*sqrt(9651062*sqrt(35) + 63
240625)*(179*sqrt(35) - 544) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) - 121835^(1/4)*sqrt(155)*sqrt(118)*(965
1062*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 63240625*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 1
2*x + 4))*sqrt(9651062*sqrt(35) + 63240625)*log(7905078125/26629*121835^(1/4)*sqrt(155)*sqrt(118)*(544*sqrt(35
)*sqrt(31) - 6265*sqrt(31))*sqrt(2*x + 1)*sqrt(9651062*sqrt(35) + 63240625) + 156873508613281250*x + 156873508
61328125*sqrt(35) + 78436754306640625) + 121835^(1/4)*sqrt(155)*sqrt(118)*(9651062*sqrt(35)*sqrt(31)*(25*x^4 +
30*x^3 + 29*x^2 + 12*x + 4) - 63240625*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt(9651062*sqrt(35)
+ 63240625)*log(-7905078125/26629*121835^(1/4)*sqrt(155)*sqrt(118)*(544*sqrt(35)*sqrt(31) - 6265*sqrt(31))*sqr
t(2*x + 1)*sqrt(9651062*sqrt(35) + 63240625) + 156873508613281250*x + 15687350861328125*sqrt(35) + 78436754306
640625) - 16381738730350*(5440*x^3 - 3629*x^2 - 4167*x - 2689)*sqrt(2*x + 1))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x
+ 4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)**(7/2)/(5*x**2+3*x+2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (213) = 426\).
time = 1.37, size = 642, normalized size = 2.14 \begin {gather*} \frac {1}{89410238750} \, \sqrt {31} {\left (28560 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 136 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 272 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 57120 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 1534925 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 3069850 \, \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}{89410238750} \, \sqrt {31} {\left (28560 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 136 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 272 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 57120 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 1534925 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 3069850 \, \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}{178820477500} \, \sqrt {31} {\left (136 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 28560 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 57120 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 272 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 1534925 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 3069850 \, \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}{178820477500} \, \sqrt {31} {\left (136 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 28560 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 57120 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 272 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 1534925 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 3069850 \, \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 {2 \, {\left (2720 \, {\left (2 \, x + 1\right )}^{\frac {7}{2}} - 11789 \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} + 7084 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} - 8771 \, \sqrt {2 \, x + 1}\right )}}{4805 \, {\left (5 \, {\left (2 \, x + 1\right )}^{2} - 8 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")

[Out]

1/89410238750*sqrt(31)*(28560*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 136*sqrt(31)
*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 272*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 57120*(7/5)^(3/4)*sq
rt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 1534925*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 3069850*
(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)) + 1/89410238750*sqrt(31)*(28560*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqr
t(-140*sqrt(35) + 2450) - 136*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 272*(7/5)^(3/4)*(140*sqrt(35
) + 2450)^(3/2) + 57120*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 1534925*sqrt(31)*(7/5)^(1/4)
*sqrt(-140*sqrt(35) + 2450) + 3069850*(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)) + 1/178820477500*sqrt(31)*(136*sqrt
(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 28560*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35
) - 35) - 57120*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 272*(7/5)^(3/4)*(-140*sqrt(35) + 24
50)^(3/2) + 1534925*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 3069850*(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) - 1/178820477500*sqrt(
31)*(136*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 28560*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 245
0)*(2*sqrt(35) - 35) - 57120*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 272*(7/5)^(3/4)*(-140*
sqrt(35) + 2450)^(3/2) + 1534925*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 3069850*(7/5)^(1/4)*sqrt(-14
0*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) + 2/4805
*(2720*(2*x + 1)^(7/2) - 11789*(2*x + 1)^(5/2) + 7084*(2*x + 1)^(3/2) - 8771*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8
*x + 3)^2

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Mupad [B]
time = 1.06, size = 245, normalized size = 0.82 \begin {gather*} \frac {\frac {17542\,\sqrt {2\,x+1}}{120125}-\frac {14168\,{\left (2\,x+1\right )}^{3/2}}{120125}+\frac {23578\,{\left (2\,x+1\right )}^{5/2}}{120125}-\frac {1088\,{\left (2\,x+1\right )}^{7/2}}{24025}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{360750390625\,\left (\frac {86779616}{72150078125}+\frac {\sqrt {31}\,17221232{}\mathrm {i}}{72150078125}\right )}+\frac {27488\,\sqrt {31}\,\sqrt {155}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{11183262109375\,\left (\frac {86779616}{72150078125}+\frac {\sqrt {31}\,17221232{}\mathrm {i}}{72150078125}\right )}\right )\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{744775}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{360750390625\,\left (-\frac {86779616}{72150078125}+\frac {\sqrt {31}\,17221232{}\mathrm {i}}{72150078125}\right )}-\frac {27488\,\sqrt {31}\,\sqrt {155}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{11183262109375\,\left (-\frac {86779616}{72150078125}+\frac {\sqrt {31}\,17221232{}\mathrm {i}}{72150078125}\right )}\right )\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{744775} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 1)^(7/2)/(3*x + 5*x^2 + 2)^3,x)

[Out]

((17542*(2*x + 1)^(1/2))/120125 - (14168*(2*x + 1)^(3/2))/120125 + (23578*(2*x + 1)^(5/2))/120125 - (1088*(2*x
+ 1)^(7/2))/24025)/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) + (155^(1/2)*a
tan((155^(1/2)*(- 31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2)*13744i)/(360750390625*((31^(1/2)*17221232i
)/72150078125 + 86779616/72150078125)) + (27488*31^(1/2)*155^(1/2)*(- 31^(1/2)*825499i - 9651062)^(1/2)*(2*x +
1)^(1/2))/(11183262109375*((31^(1/2)*17221232i)/72150078125 + 86779616/72150078125)))*(- 31^(1/2)*825499i - 9
651062)^(1/2)*1i)/744775 - (155^(1/2)*atan((155^(1/2)*(31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2)*13744
i)/(360750390625*((31^(1/2)*17221232i)/72150078125 - 86779616/72150078125)) - (27488*31^(1/2)*155^(1/2)*(31^(1
/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2))/(11183262109375*((31^(1/2)*17221232i)/72150078125 - 86779616/721
50078125)))*(31^(1/2)*825499i - 9651062)^(1/2)*1i)/744775

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