3.850 \(\int x^3 (2+3 x)^{3/2} \sqrt{1+4 x} \, dx\)

Optimal. Leaf size=146 \[ \frac{1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}+\frac{(4103-7968 x) (4 x+1)^{3/2} (3 x+2)^{5/2}}{829440}-\frac{8543 \sqrt{4 x+1} (3 x+2)^{5/2}}{995328}+\frac{42715 \sqrt{4 x+1} (3 x+2)^{3/2}}{15925248}+\frac{213575 \sqrt{4 x+1} \sqrt{3 x+2}}{42467328}+\frac{1067875 \sinh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{4 x+1}\right )}{84934656 \sqrt{3}} \]

[Out]

(213575*Sqrt[2 + 3*x]*Sqrt[1 + 4*x])/42467328 + (42715*(2 + 3*x)^(3/2)*Sqrt[1 + 4*x])/15925248 - (8543*(2 + 3*
x)^(5/2)*Sqrt[1 + 4*x])/995328 + ((4103 - 7968*x)*(2 + 3*x)^(5/2)*(1 + 4*x)^(3/2))/829440 + (x^2*(2 + 3*x)^(5/
2)*(1 + 4*x)^(3/2))/72 + (1067875*ArcSinh[Sqrt[3/5]*Sqrt[1 + 4*x]])/(84934656*Sqrt[3])

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Rubi [A]  time = 0.0389412, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {100, 147, 50, 54, 215} \[ \frac{1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}+\frac{(4103-7968 x) (4 x+1)^{3/2} (3 x+2)^{5/2}}{829440}-\frac{8543 \sqrt{4 x+1} (3 x+2)^{5/2}}{995328}+\frac{42715 \sqrt{4 x+1} (3 x+2)^{3/2}}{15925248}+\frac{213575 \sqrt{4 x+1} \sqrt{3 x+2}}{42467328}+\frac{1067875 \sinh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{4 x+1}\right )}{84934656 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(213575*Sqrt[2 + 3*x]*Sqrt[1 + 4*x])/42467328 + (42715*(2 + 3*x)^(3/2)*Sqrt[1 + 4*x])/15925248 - (8543*(2 + 3*
x)^(5/2)*Sqrt[1 + 4*x])/995328 + ((4103 - 7968*x)*(2 + 3*x)^(5/2)*(1 + 4*x)^(3/2))/829440 + (x^2*(2 + 3*x)^(5/
2)*(1 + 4*x)^(3/2))/72 + (1067875*ArcSinh[Sqrt[3/5]*Sqrt[1 + 4*x]])/(84934656*Sqrt[3])

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 147

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

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

Rubi steps

\begin{align*} \int x^3 (2+3 x)^{3/2} \sqrt{1+4 x} \, dx &=\frac{1}{72} x^2 (2+3 x)^{5/2} (1+4 x)^{3/2}+\frac{1}{72} \int \left (-4-\frac{83 x}{2}\right ) x (2+3 x)^{3/2} \sqrt{1+4 x} \, dx\\ &=\frac{(4103-7968 x) (2+3 x)^{5/2} (1+4 x)^{3/2}}{829440}+\frac{1}{72} x^2 (2+3 x)^{5/2} (1+4 x)^{3/2}-\frac{8543 \int (2+3 x)^{3/2} \sqrt{1+4 x} \, dx}{110592}\\ &=-\frac{8543 (2+3 x)^{5/2} \sqrt{1+4 x}}{995328}+\frac{(4103-7968 x) (2+3 x)^{5/2} (1+4 x)^{3/2}}{829440}+\frac{1}{72} x^2 (2+3 x)^{5/2} (1+4 x)^{3/2}+\frac{42715 \int \frac{(2+3 x)^{3/2}}{\sqrt{1+4 x}} \, dx}{1990656}\\ &=\frac{42715 (2+3 x)^{3/2} \sqrt{1+4 x}}{15925248}-\frac{8543 (2+3 x)^{5/2} \sqrt{1+4 x}}{995328}+\frac{(4103-7968 x) (2+3 x)^{5/2} (1+4 x)^{3/2}}{829440}+\frac{1}{72} x^2 (2+3 x)^{5/2} (1+4 x)^{3/2}+\frac{213575 \int \frac{\sqrt{2+3 x}}{\sqrt{1+4 x}} \, dx}{10616832}\\ &=\frac{213575 \sqrt{2+3 x} \sqrt{1+4 x}}{42467328}+\frac{42715 (2+3 x)^{3/2} \sqrt{1+4 x}}{15925248}-\frac{8543 (2+3 x)^{5/2} \sqrt{1+4 x}}{995328}+\frac{(4103-7968 x) (2+3 x)^{5/2} (1+4 x)^{3/2}}{829440}+\frac{1}{72} x^2 (2+3 x)^{5/2} (1+4 x)^{3/2}+\frac{1067875 \int \frac{1}{\sqrt{2+3 x} \sqrt{1+4 x}} \, dx}{84934656}\\ &=\frac{213575 \sqrt{2+3 x} \sqrt{1+4 x}}{42467328}+\frac{42715 (2+3 x)^{3/2} \sqrt{1+4 x}}{15925248}-\frac{8543 (2+3 x)^{5/2} \sqrt{1+4 x}}{995328}+\frac{(4103-7968 x) (2+3 x)^{5/2} (1+4 x)^{3/2}}{829440}+\frac{1}{72} x^2 (2+3 x)^{5/2} (1+4 x)^{3/2}+\frac{1067875 \operatorname{Subst}\left (\int \frac{1}{\sqrt{5+3 x^2}} \, dx,x,\sqrt{1+4 x}\right )}{84934656}\\ &=\frac{213575 \sqrt{2+3 x} \sqrt{1+4 x}}{42467328}+\frac{42715 (2+3 x)^{3/2} \sqrt{1+4 x}}{15925248}-\frac{8543 (2+3 x)^{5/2} \sqrt{1+4 x}}{995328}+\frac{(4103-7968 x) (2+3 x)^{5/2} (1+4 x)^{3/2}}{829440}+\frac{1}{72} x^2 (2+3 x)^{5/2} (1+4 x)^{3/2}+\frac{1067875 \sinh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{1+4 x}\right )}{84934656 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.140451, size = 75, normalized size = 0.51 \[ \frac{6 \sqrt{3 x+2} \sqrt{4 x+1} \left (106168320 x^5+94666752 x^4+4119552 x^3-1849728 x^2+1089592 x-881613\right )+5339375 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{4 x+1}\right )}{1274019840} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[2 + 3*x]*Sqrt[1 + 4*x]*(-881613 + 1089592*x - 1849728*x^2 + 4119552*x^3 + 94666752*x^4 + 106168320*x^5
) + 5339375*Sqrt[3]*ArcSinh[Sqrt[3/5]*Sqrt[1 + 4*x]])/1274019840

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Maple [A]  time = 0.013, size = 157, normalized size = 1.1 \begin{align*}{\frac{1}{2548039680}\sqrt{2+3\,x}\sqrt{4\,x+1} \left ( 1274019840\,{x}^{5}\sqrt{12\,{x}^{2}+11\,x+2}[phone]\,{x}^{4}\sqrt{12\,{x}^{2}+11\,x+2}[phone]\,{x}^{3}\sqrt{12\,{x}^{2}+11\,x+2}-22196736\,{x}^{2}\sqrt{12\,{x}^{2}+11\,x+2}+5339375\,\ln \left ({\frac{11\,\sqrt{3}}{12}}+2\,x\sqrt{3}+\sqrt{12\,{x}^{2}+11\,x+2} \right ) \sqrt{3}[phone]\,\sqrt{12\,{x}^{2}+11\,x+2}x-10579356\,\sqrt{12\,{x}^{2}+11\,x+2} \right ){\frac{1}{\sqrt{12\,{x}^{2}+11\,x+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2548039680*(2+3*x)^(1/2)*(4*x+1)^(1/2)*(1274019840*x^5*(12*x^2+11*x+2)^(1/2)[phone]*x^4*(12*x^2+11*x+2)^
(1/2)[phone]*x^3*(12*x^2+11*x+2)^(1/2)-22196736*x^2*(12*x^2+11*x+2)^(1/2)+5339375*ln(11/12*3^(1/2)+2*x*3^(1/
2)+(12*x^2+11*x+2)^(1/2))*3^(1/2)[phone]*(12*x^2+11*x+2)^(1/2)*x-10579356*(12*x^2+11*x+2)^(1/2))/(12*x^2+11*
x+2)^(1/2)

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Maxima [A]  time = 2.65445, size = 163, normalized size = 1.12 \begin{align*} \frac{1}{24} \,{\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac{3}{2}} x^{3} - \frac{1}{960} \,{\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac{3}{2}} x^{2} - \frac{403}{92160} \,{\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac{3}{2}} x + \frac{22933}{6635520} \,{\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac{3}{2}} - \frac{42715}{1769472} \, \sqrt{12 \, x^{2} + 11 \, x + 2} x + \frac{1067875}{509607936} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{12 \, x^{2} + 11 \, x + 2} + 24 \, x + 11\right ) - \frac{469865}{42467328} \, \sqrt{12 \, x^{2} + 11 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(12*x^2 + 11*x + 2)^(3/2)*x^3 - 1/960*(12*x^2 + 11*x + 2)^(3/2)*x^2 - 403/92160*(12*x^2 + 11*x + 2)^(3/2)
*x + 22933/6635520*(12*x^2 + 11*x + 2)^(3/2) - 42715/1769472*sqrt(12*x^2 + 11*x + 2)*x + 1067875/509607936*sqr
t(3)*log(4*sqrt(3)*sqrt(12*x^2 + 11*x + 2) + 24*x + 11) - 469865/42467328*sqrt(12*x^2 + 11*x + 2)

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Fricas [A]  time = 1.66253, size = 312, normalized size = 2.14 \begin{align*} \frac{1}{212336640} \,{\left (106168320 \, x^{5} + 94666752 \, x^{4} + 4119552 \, x^{3} - 1849728 \, x^{2} + 1089592 \, x - 881613\right )} \sqrt{4 \, x + 1} \sqrt{3 \, x + 2} + \frac{1067875}{1019215872} \, \sqrt{3} \log \left (8 \, \sqrt{3}{\left (24 \, x + 11\right )} \sqrt{4 \, x + 1} \sqrt{3 \, x + 2} + 1152 \, x^{2} + 1056 \, x + 217\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/212336640*(106168320*x^5 + 94666752*x^4 + 4119552*x^3 - 1849728*x^2 + 1089592*x - 881613)*sqrt(4*x + 1)*sqrt
(3*x + 2) + 1067875/1019215872*sqrt(3)*log(8*sqrt(3)*(24*x + 11)*sqrt(4*x + 1)*sqrt(3*x + 2) + 1152*x^2 + 1056
*x + 217)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(2+3*x)**(3/2)*(1+4*x)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 3.1315, size = 181, normalized size = 1.24 \begin{align*} \frac{1}{23592960} \,{\left (2 \,{\left (12 \,{\left (2 \,{\left (8 \,{\left (120 \, x - 109\right )}{\left (4 \, x + 1\right )} + 1845\right )}{\left (4 \, x + 1\right )} - 1415\right )}{\left (4 \, x + 1\right )} - 62545\right )}{\left (4 \, x + 1\right )} + 427925\right )} \sqrt{4 \, x + 1} \sqrt{3 \, x + 2} + \frac{1}{6635520} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (96 \, x - 61\right )}{\left (4 \, x + 1\right )} + 1535\right )}{\left (4 \, x + 1\right )} + 13465\right )}{\left (4 \, x + 1\right )} - 153725\right )} \sqrt{4 \, x + 1} \sqrt{3 \, x + 2} - \frac{1067875}{254803968} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{4 \, x + 1} + 2 \, \sqrt{3 \, x + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/23592960*(2*(12*(2*(8*(120*x - 109)*(4*x + 1) + 1845)*(4*x + 1) - 1415)*(4*x + 1) - 62545)*(4*x + 1) + 42792
5)*sqrt(4*x + 1)*sqrt(3*x + 2) + 1/6635520*(2*(12*(18*(96*x - 61)*(4*x + 1) + 1535)*(4*x + 1) + 13465)*(4*x +
1) - 153725)*sqrt(4*x + 1)*sqrt(3*x + 2) - 1067875/254803968*sqrt(3)*log(-sqrt(3)*sqrt(4*x + 1) + 2*sqrt(3*x +
2))