3.24.90 \(\int \frac {(2+3 x)^4 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=161 \[ \frac {(5 x+3)^{3/2} (3 x+2)^4}{\sqrt {1-2 x}}+\frac {33}{20} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^3+\frac {10377 \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{1600}+\frac {9 \sqrt {1-2 x} (5 x+3)^{3/2} (2253560 x+4772357)}{256000}+\frac {1018114917 \sqrt {1-2 x} \sqrt {5 x+3}}{1024000}-\frac {11199264087 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1024000 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 153, 147, 50, 54, 216} \begin {gather*} \frac {(5 x+3)^{3/2} (3 x+2)^4}{\sqrt {1-2 x}}+\frac {33}{20} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^3+\frac {10377 \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{1600}+\frac {9 \sqrt {1-2 x} (5 x+3)^{3/2} (2253560 x+4772357)}{256000}+\frac {1018114917 \sqrt {1-2 x} \sqrt {5 x+3}}{1024000}-\frac {11199264087 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1024000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1018114917*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1024000 + (10377*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/1600 + (3
3*Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 + ((2 + 3*x)^4*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (9*Sqrt[1 - 2*
x]*(3 + 5*x)^(3/2)*(4772357 + 2253560*x))/256000 - (11199264087*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1024000*Sqr
t[10])

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 97

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

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 153

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

Rule 216

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

Rubi steps

\begin {align*} \int \frac {(2+3 x)^4 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^3 \sqrt {3+5 x} \left (51+\frac {165 x}{2}\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1}{50} \int \frac {\left (-8070-\frac {51885 x}{4}\right ) (2+3 x)^2 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {\int \frac {(2+3 x) \sqrt {3+5 x} \left (\frac {3983295}{4}+\frac {12676275 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{2000}\\ &=\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac {1018114917 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{512000}\\ &=\frac {1018114917 \sqrt {1-2 x} \sqrt {3+5 x}}{1024000}+\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac {11199264087 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2048000}\\ &=\frac {1018114917 \sqrt {1-2 x} \sqrt {3+5 x}}{1024000}+\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac {11199264087 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1024000 \sqrt {5}}\\ &=\frac {1018114917 \sqrt {1-2 x} \sqrt {3+5 x}}{1024000}+\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac {11199264087 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1024000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 93, normalized size = 0.58 \begin {gather*} \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (41472000 x^5+200966400 x^4+461171520 x^3+732415080 x^2+1206337246 x-1702927233\right )-11199264087 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{10240000 \sqrt {-(1-2 x)^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-1702927233 + 1206337246*x + 732415080*x^2 + 461171520*x^3 + 200966400*x^4
+ 41472000*x^5) - 11199264087*Sqrt[10]*(-1 + 2*x)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(10240000*Sqrt[-(1 - 2*x
)^2])

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IntegrateAlgebraic [A]  time = 0.81, size = 149, normalized size = 0.93 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (331776 (5 x+3)^{11/2}+3062016 (5 x+3)^{9/2}[phone] (5 x+3)^{7/2}[phone] (5 x+3)^{5/2}[phone] (5 x+3)^{3/2}-55996320435 \sqrt {5 x+3}\right )}{5120000 \sqrt {5} (2 (5 x+3)-11)}+\frac {11199264087 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{512000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-55996320435*Sqrt[3 + 5*x] + 3393716390*(3 + 5*x)^(3/2) + 246814248*(3 + 5*x)^(5/2) +
25630272*(3 + 5*x)^(7/2) + 3062016*(3 + 5*x)^(9/2) + 331776*(3 + 5*x)^(11/2)))/(5120000*Sqrt[5]*(-11 + 2*(3 +
5*x))) + (11199264087*ArcTan[(Sqrt[2]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(512000*Sqrt[10])

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fricas [A]  time = 1.29, size = 96, normalized size = 0.60 \begin {gather*} \frac {11199264087 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (41472000 \, x^{5} + 200966400 \, x^{4} + 461171520 \, x^{3} + 732415080 \, x^{2} + 1206337246 \, x - 1702927233\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20480000 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20480000*(11199264087*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^
2 + x - 3)) + 20*(41472000*x^5 + 200966400*x^4 + 461171520*x^3 + 732415080*x^2 + 1206337246*x - 1702927233)*sq
rt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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giac [A]  time = 1.19, size = 110, normalized size = 0.68 \begin {gather*} -\frac {11199264087}{10240000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (12 \, {\left (24 \, {\left (12 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} + 443 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 44497 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 10283927 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1696858195 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 55996320435 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{128000000 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11199264087/10240000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/128000000*(2*(12*(24*(12*(48*sqrt(5)*(5
*x + 3) + 443*sqrt(5))*(5*x + 3) + 44497*sqrt(5))*(5*x + 3) + 10283927*sqrt(5))*(5*x + 3) + 1696858195*sqrt(5)
)*(5*x + 3) - 55996320435*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

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maple [A]  time = 0.01, size = 157, normalized size = 0.98 \begin {gather*} -\frac {\left (-829440000 \sqrt {-10 x^{2}-x +3}\, x^{5}-4019328000 \sqrt {-10 x^{2}-x +3}\, x^{4}-9223430400 \sqrt {-10 x^{2}-x +3}\, x^{3}-14648301600 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-24126744920 \sqrt {-10 x^{2}-x +3}\, x -11199264087 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{20480000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20480000*(-829440000*(-10*x^2-x+3)^(1/2)*x^5-4019328000*(-10*x^2-x+3)^(1/2)*x^4-9223430400*(-10*x^2-x+3)^(1
/2)*x^3+22398528174*10^(1/2)*x*arcsin(20/11*x+1/11)-14648301600*(-10*x^2-x+3)^(1/2)*x^2-11199264087*10^(1/2)*a
rcsin(20/11*x+1/11)-24126744920*(-10*x^2-x+3)^(1/2)*x+34058544660*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^
(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [C]  time = 1.29, size = 198, normalized size = 1.23 \begin {gather*} \frac {81}{400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} - \frac {6669}{640} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {12607994487}{20480000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1760913}{25600} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) - \frac {359469}{12800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {14553}{64} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x - \frac {2420847}{51200} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {305613}{1280} \, \sqrt {10 \, x^{2} - 21 \, x + 8} + \frac {540891153}{1024000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2401 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {1029 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{16 \, {\left (2 \, x - 1\right )}} - \frac {79233 \, \sqrt {-10 \, x^{2} - x + 3}}{64 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/400*(-10*x^2 - x + 3)^(5/2) - 6669/640*(-10*x^2 - x + 3)^(3/2)*x - 12607994487/20480000*sqrt(5)*sqrt(2)*arc
sin(20/11*x + 1/11) - 1760913/25600*I*sqrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) - 359469/12800*(-10*x^2 - x + 3)
^(3/2) + 14553/64*sqrt(10*x^2 - 21*x + 8)*x - 2420847/51200*sqrt(-10*x^2 - x + 3)*x - 305613/1280*sqrt(10*x^2
- 21*x + 8) + 540891153/1024000*sqrt(-10*x^2 - x + 3) - 2401/32*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) - 10
29/16*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) - 79233/64*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  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((2+3*x)**4*(3+5*x)**(3/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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