3.2.34 \(\int \frac {1+3 x+4 x^2}{(1+2 x)^2 (2+3 x^2)^{5/2}} \, dx\) [134]

Optimal. Leaf size=95 \[ \frac {-10+97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac {24+887 x}{7986 \sqrt {2+3 x^2}}-\frac {16 \sqrt {2+3 x^2}}{1331 (1+2 x)}-\frac {32 \tanh ^{-1}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{1331 \sqrt {11}} \]

[Out]

1/726*(-10+97*x)/(3*x^2+2)^(3/2)-32/14641*arctanh(1/11*(4-3*x)*11^(1/2)/(3*x^2+2)^(1/2))*11^(1/2)+1/7986*(24+8
87*x)/(3*x^2+2)^(1/2)-16/1331*(3*x^2+2)^(1/2)/(1+2*x)

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Rubi [A]
time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1661, 821, 739, 212} \begin {gather*} -\frac {10-97 x}{726 \left (3 x^2+2\right )^{3/2}}-\frac {16 \sqrt {3 x^2+2}}{1331 (2 x+1)}+\frac {887 x+24}{7986 \sqrt {3 x^2+2}}-\frac {32 \tanh ^{-1}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{1331 \sqrt {11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 + 3*x^2)^(5/2)),x]

[Out]

-1/726*(10 - 97*x)/(2 + 3*x^2)^(3/2) + (24 + 887*x)/(7986*Sqrt[2 + 3*x^2]) - (16*Sqrt[2 + 3*x^2])/(1331*(1 + 2
*x)) - (32*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/(1331*Sqrt[11])

Rule 212

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 821

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

Rule 1661

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[(a*g - c*f*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx &=-\frac {10-97 x}{726 \left (2+3 x^2\right )^{3/2}}-\frac {1}{18} \int \frac {-\frac {798}{121}-\frac {1968 x}{121}-\frac {2328 x^2}{121}}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {10-97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac {24+887 x}{7986 \sqrt {2+3 x^2}}+\frac {1}{108} \int \frac {\frac {10368}{1331}+\frac {1728 x}{1331}}{(1+2 x)^2 \sqrt {2+3 x^2}} \, dx\\ &=-\frac {10-97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac {24+887 x}{7986 \sqrt {2+3 x^2}}-\frac {16 \sqrt {2+3 x^2}}{1331 (1+2 x)}+\frac {32 \int \frac {1}{(1+2 x) \sqrt {2+3 x^2}} \, dx}{1331}\\ &=-\frac {10-97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac {24+887 x}{7986 \sqrt {2+3 x^2}}-\frac {16 \sqrt {2+3 x^2}}{1331 (1+2 x)}-\frac {32 \text {Subst}\left (\int \frac {1}{11-x^2} \, dx,x,\frac {4-3 x}{\sqrt {2+3 x^2}}\right )}{1331}\\ &=-\frac {10-97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac {24+887 x}{7986 \sqrt {2+3 x^2}}-\frac {16 \sqrt {2+3 x^2}}{1331 (1+2 x)}-\frac {32 \tanh ^{-1}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{1331 \sqrt {11}}\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 91, normalized size = 0.96 \begin {gather*} \frac {11 \left (-446+2717 x+4602 x^2+2805 x^3+4458 x^4\right )-192 \sqrt {22+33 x^2} \left (2+4 x+3 x^2+6 x^3\right ) \tanh ^{-1}\left (\frac {4-3 x}{\sqrt {22+33 x^2}}\right )}{87846 (1+2 x) \left (2+3 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 + 3*x^2)^(5/2)),x]

[Out]

(11*(-446 + 2717*x + 4602*x^2 + 2805*x^3 + 4458*x^4) - 192*Sqrt[22 + 33*x^2]*(2 + 4*x + 3*x^2 + 6*x^3)*ArcTanh
[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(87846*(1 + 2*x)*(2 + 3*x^2)^(3/2))

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Maple [A]
time = 0.09, size = 143, normalized size = 1.51

method	result	size
risch	\(\frac {4458 x^{4}+2805 x^{3}+4602 x^{2}+2717 x -446}{7986 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (2 x +1\right )}-\frac {32 \sqrt {11}\, \arctanh \left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-12 x +5}}\right )}{14641}\)	\(70\)
trager	\(\frac {4458 x^{4}+2805 x^{3}+4602 x^{2}+2717 x -446}{7986 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (2 x +1\right )}+\frac {32 \RootOf \left (\textit {\_Z}^{2}-11\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-11\right ) x +11 \sqrt {3 x^{2}+2}-4 \RootOf \left (\textit {\_Z}^{2}-11\right )}{2 x +1}\right )}{14641}\)	\(86\)
default	\(\frac {x}{6 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {x}{6 \sqrt {3 x^{2}+2}}+\frac {4}{363 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}-\frac {10 x}{121 \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}-\frac {98 x}{1331 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}+\frac {16}{1331 \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}}}-\frac {32 \sqrt {11}\, \arctanh \left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-12 x +5}}\right )}{14641}-\frac {1}{22 \left (x +\frac {1}{2}\right ) \left (3 \left (x +\frac {1}{2}\right )^{2}-3 x +\frac {5}{4}\right )^{\frac {3}{2}}}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+3*x+1)/(2*x+1)^2/(3*x^2+2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6/(3*x^2+2)^(3/2)*x+1/6*x/(3*x^2+2)^(1/2)+4/363/(3*(x+1/2)^2-3*x+5/4)^(3/2)-10/121*x/(3*(x+1/2)^2-3*x+5/4)^(
3/2)-98/1331*x/(3*(x+1/2)^2-3*x+5/4)^(1/2)+16/1331/(3*(x+1/2)^2-3*x+5/4)^(1/2)-32/14641*11^(1/2)*arctanh(2/11*
(4-3*x)*11^(1/2)/(12*(x+1/2)^2-12*x+5)^(1/2))-1/22/(x+1/2)/(3*(x+1/2)^2-3*x+5/4)^(3/2)

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Maxima [A]
time = 0.50, size = 107, normalized size = 1.13 \begin {gather*} \frac {32}{14641} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {743 \, x}{7986 \, \sqrt {3 \, x^{2} + 2}} + \frac {16}{1331 \, \sqrt {3 \, x^{2} + 2}} + \frac {61 \, x}{726 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {1}{11 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {4}{363 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

32/14641*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs(2*x + 1)) + 743/7986*x/sqrt(3*x^2 + 2)
+ 16/1331/sqrt(3*x^2 + 2) + 61/726*x/(3*x^2 + 2)^(3/2) - 1/11/(2*(3*x^2 + 2)^(3/2)*x + (3*x^2 + 2)^(3/2)) + 4/
363/(3*x^2 + 2)^(3/2)

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Fricas [A]
time = 0.36, size = 134, normalized size = 1.41 \begin {gather*} \frac {96 \, \sqrt {11} {\left (18 \, x^{5} + 9 \, x^{4} + 24 \, x^{3} + 12 \, x^{2} + 8 \, x + 4\right )} \log \left (-\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, {\left (4458 \, x^{4} + 2805 \, x^{3} + 4602 \, x^{2} + 2717 \, x - 446\right )} \sqrt {3 \, x^{2} + 2}}{87846 \, {\left (18 \, x^{5} + 9 \, x^{4} + 24 \, x^{3} + 12 \, x^{2} + 8 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/87846*(96*sqrt(11)*(18*x^5 + 9*x^4 + 24*x^3 + 12*x^2 + 8*x + 4)*log(-(sqrt(11)*sqrt(3*x^2 + 2)*(3*x - 4) + 2
1*x^2 - 12*x + 19)/(4*x^2 + 4*x + 1)) + 11*(4458*x^4 + 2805*x^3 + 4602*x^2 + 2717*x - 446)*sqrt(3*x^2 + 2))/(1
8*x^5 + 9*x^4 + 24*x^3 + 12*x^2 + 8*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((4*x**2+3*x+1)/(1+2*x)**2/(3*x**2+2)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (76) = 152\).
time = 5.20, size = 233, normalized size = 2.45 \begin {gather*} -\frac {1}{263538} \, \sqrt {11} {\left (743 \, \sqrt {11} \sqrt {3} - 576 \, \log \left (\sqrt {11} \sqrt {3} - 3\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {32 \, \sqrt {11} \log \left (\sqrt {11} {\left (\sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {11}}{2 \, x + 1}\right )} - 3\right )}{14641 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {\frac {\frac {\frac {11 \, {\left (\frac {731}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {528}{{\left (2 \, x + 1\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}\right )}}{2 \, x + 1} - \frac {14163}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} + \frac {6111}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} - \frac {2229}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{7986 \, {\left (\frac {6}{2 \, x + 1} - \frac {11}{{\left (2 \, x + 1\right )}^{2}} - 3\right )} \sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

-1/263538*sqrt(11)*(743*sqrt(11)*sqrt(3) - 576*log(sqrt(11)*sqrt(3) - 3))*sgn(1/(2*x + 1)) - 32/14641*sqrt(11)
*log(sqrt(11)*(sqrt(-6/(2*x + 1) + 11/(2*x + 1)^2 + 3) + sqrt(11)/(2*x + 1)) - 3)/sgn(1/(2*x + 1)) + 1/7986*((
(11*(731/sgn(1/(2*x + 1)) + 528/((2*x + 1)*sgn(1/(2*x + 1))))/(2*x + 1) - 14163/sgn(1/(2*x + 1)))/(2*x + 1) +
6111/sgn(1/(2*x + 1)))/(2*x + 1) - 2229/sgn(1/(2*x + 1)))/((6/(2*x + 1) - 11/(2*x + 1)^2 - 3)*sqrt(-6/(2*x + 1
) + 11/(2*x + 1)^2 + 3))

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Mupad [B]
time = 4.31, size = 270, normalized size = 2.84 \begin {gather*} \frac {\sqrt {11}\,\left (8\,\ln \left (x+\frac {1}{2}\right )-8\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )\right )}{14641}+\frac {\sqrt {11}\,\left (\frac {48\,\ln \left (x+\frac {1}{2}\right )}{1331}-\frac {48\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )}{1331}\right )}{22}-\frac {8\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1331\,\left (x+\frac {1}{2}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {291}{1936}+\frac {\sqrt {6}\,15{}\mathrm {i}}{1936}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {97}{968}+\frac {\sqrt {6}\,5{}\mathrm {i}}{968}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {291}{1936}+\frac {\sqrt {6}\,15{}\mathrm {i}}{1936}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {97}{968}+\frac {\sqrt {6}\,5{}\mathrm {i}}{968}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-288+\sqrt {6}\,2481{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1149984\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (288+\sqrt {6}\,2481{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1149984\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 4*x^2 + 1)/((2*x + 1)^2*(3*x^2 + 2)^(5/2)),x)

[Out]

(11^(1/2)*(8*log(x + 1/2) - 8*log(x - (3^(1/2)*11^(1/2)*(x^2 + 2/3)^(1/2))/3 - 4/3)))/14641 + (11^(1/2)*((48*l
og(x + 1/2))/1331 - (48*log(x - (3^(1/2)*11^(1/2)*(x^2 + 2/3)^(1/2))/3 - 4/3))/1331))/22 - (8*3^(1/2)*(x^2 + 2
/3)^(1/2))/(1331*(x + 1/2)) - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*15i)/1936 - 291/1936)/(x + (6^(1/2)*1i)/3)
+ (6^(1/2)*((6^(1/2)*5i)/968 - 97/968)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^
(1/2)*15i)/1936 + 291/1936)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*5i)/968 + 97/968)*1i)/(2*(x - (6^(1/2)*1
i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*2481i - 288)*(x^2 + 2/3)^(1/2)*1i)/(1149984*(x + (6^(1/2)*1i)/3)) -
(3^(1/2)*6^(1/2)*(6^(1/2)*2481i + 288)*(x^2 + 2/3)^(1/2)*1i)/(1149984*(x - (6^(1/2)*1i)/3))

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