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

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

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Rubi [A]  time = 0.04, antiderivative size = 146, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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 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 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*}

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Mathematica [A]  time = 0.16, size = 75, normalized size = 0.51 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.38, size = 158, normalized size = 1.08 \begin {gather*} \frac {1067875 \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {4 x+1}}{2 \sqrt {3 x+2}}\right )}{84934656 \sqrt {3}}-\frac {125 \sqrt {4 x+1} \left (\frac {2075949 (4 x+1)^5}{(3 x+2)^5}-\frac {15684948 (4 x+1)^4}{(3 x+2)^4}+\frac {3417120 (4 x+1)^3}{(3 x+2)^3}+\frac {29918592 (4 x+1)^2}{(3 x+2)^2}-\frac {30101248 (4 x+1)}{3 x+2}+8748032\right )}{42467328 \sqrt {3 x+2} \left (\frac {3 (4 x+1)}{3 x+2}-4\right )^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-125*Sqrt[1 + 4*x]*(8748032 - (30101248*(1 + 4*x))/(2 + 3*x) + (29918592*(1 + 4*x)^2)/(2 + 3*x)^2 + (3417120*
(1 + 4*x)^3)/(2 + 3*x)^3 - (15684948*(1 + 4*x)^4)/(2 + 3*x)^4 + (2075949*(1 + 4*x)^5)/(2 + 3*x)^5))/(42467328*
Sqrt[2 + 3*x]*(-4 + (3*(1 + 4*x))/(2 + 3*x))^6) + (1067875*ArcTanh[(Sqrt[3]*Sqrt[1 + 4*x])/(2*Sqrt[2 + 3*x])])
/(84934656*Sqrt[3])

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fricas [A]  time = 1.18, size = 82, normalized size = 0.56 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.40, size = 173, normalized size = 1.18 \begin {gather*} \frac {1}{14155776} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (24 \, x - 29\right )} {\left (4 \, x + 1\right )} + 645\right )} {\left (4 \, x + 1\right )} - 3685\right )} {\left (4 \, x + 1\right )} - 28835\right )} {\left (4 \, x + 1\right )} + 448303\right )} \sqrt {4 \, x + 1} \sqrt {3 \, x + 2} + \frac {11}{53084160} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (96 \, x - 91\right )} {\left (4 \, x + 1\right )} + 3545\right )} {\left (4 \, x + 1\right )} - 10865\right )} {\left (4 \, x + 1\right )} - 239435\right )} \sqrt {4 \, x + 1} \sqrt {3 \, x + 2} + \frac {1}{221184} \, {\left (2 \, {\left (12 \, {\left (72 \, x - 49\right )} {\left (4 \, x + 1\right )} + 811\right )} {\left (4 \, x + 1\right )} + 2857\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 {gather*}

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]

1/14155776*(2*(12*(6*(8*(24*x - 29)*(4*x + 1) + 645)*(4*x + 1) - 3685)*(4*x + 1) - 28835)*(4*x + 1) + 448303)*
sqrt(4*x + 1)*sqrt(3*x + 2) + 11/53084160*(2*(12*(18*(96*x - 91)*(4*x + 1) + 3545)*(4*x + 1) - 10865)*(4*x + 1
) - 239435)*sqrt(4*x + 1)*sqrt(3*x + 2) + 1/221184*(2*(12*(72*x - 49)*(4*x + 1) + 811)*(4*x + 1) + 2857)*sqrt(
4*x + 1)*sqrt(3*x + 2) - 1067875/254803968*sqrt(3)*log(-sqrt(3)*sqrt(4*x + 1) + 2*sqrt(3*x + 2))

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maple [A]  time = 0.01, size = 157, normalized size = 1.08 \begin {gather*} \frac {\sqrt {3 x +2}\, \sqrt {4 x +1}\, \left (1274019840 \sqrt {12 x^{2}+11 x +2}\, x^{5}[phone] \sqrt {12 x^{2}+11 x +2}\, x^{4}[phone] \sqrt {12 x^{2}+11 x +2}\, x^{3}-22196736 \sqrt {12 x^{2}+11 x +2}\, x^{2}[phone] \sqrt {12 x^{2}+11 x +2}\, x +5339375 \sqrt {3}\, \ln \left (2 \sqrt {3}\, x +\frac {11 \sqrt {3}}{12}+\sqrt {12 x^{2}+11 x +2}\right )-10579356 \sqrt {12 x^{2}+11 x +2}\right )}{2548039680 \sqrt {12 x^{2}+11 x +2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2548039680*(3*x+2)^(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*3^(1/2)
*x+(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 = 1.32, size = 121, normalized size = 0.83 \begin {gather*} \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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {4\,x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 120.39, size = 167, normalized size = 1.14 \begin {gather*} \frac {3 \left (4 x + 1\right )^{\frac {13}{2}}}{4096 \sqrt {12 x + 8}} + \frac {7 \left (4 x + 1\right )^{\frac {11}{2}}}{40960 \sqrt {12 x + 8}} - \frac {869 \left (4 x + 1\right )^{\frac {9}{2}}}{196608 \sqrt {12 x + 8}} + \frac {2027 \left (4 x + 1\right )^{\frac {7}{2}}}{1179648 \sqrt {12 x + 8}} + \frac {119135 \left (4 x + 1\right )^{\frac {5}{2}}}{14155776 \sqrt {12 x + 8}} - \frac {904775 \left (4 x + 1\right )^{\frac {3}{2}}}{84934656 \sqrt {12 x + 8}} - \frac {1067875 \sqrt {4 x + 1}}{84934656 \sqrt {12 x + 8}} + \frac {1067875 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {15} \sqrt {4 x + 1}}{5} \right )}}{254803968} \end {gather*}

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]

3*(4*x + 1)**(13/2)/(4096*sqrt(12*x + 8)) + 7*(4*x + 1)**(11/2)/(40960*sqrt(12*x + 8)) - 869*(4*x + 1)**(9/2)/
(196608*sqrt(12*x + 8)) + 2027*(4*x + 1)**(7/2)/(1179648*sqrt(12*x + 8)) + 119135*(4*x + 1)**(5/2)/(14155776*s
qrt(12*x + 8)) - 904775*(4*x + 1)**(3/2)/(84934656*sqrt(12*x + 8)) - 1067875*sqrt(4*x + 1)/(84934656*sqrt(12*x
+ 8)) + 1067875*sqrt(3)*asinh(sqrt(15)*sqrt(4*x + 1)/5)/254803968

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