3.2540 \(\int \frac{(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=154 \[ \frac{(5 x+3)^{5/2} (3 x+2)^3}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac{9748787 \sqrt{1-2 x} (5 x+3)^{3/2}}{51200}+\frac{9 \sqrt{1-2 x} (5 x+3)^{5/2} (13820 x+27937)}{6400}+\frac{321709971 \sqrt{1-2 x} \sqrt{5 x+3}}{204800}-\frac{3538809681 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{204800 \sqrt{10}} \]

[Out]

(321709971*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/204800 + (9748787*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/51200 + (33*Sqrt[1 -
2*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/20 + ((2 + 3*x)^3*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (9*Sqrt[1 - 2*x]*(3 + 5*x
)^(5/2)*(27937 + 13820*x))/6400 - (3538809681*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(204800*Sqrt[10])

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Rubi [A]  time = 0.0398047, antiderivative size = 154, normalized size of antiderivative = 1., 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} \[ \frac{(5 x+3)^{5/2} (3 x+2)^3}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac{9748787 \sqrt{1-2 x} (5 x+3)^{3/2}}{51200}+\frac{9 \sqrt{1-2 x} (5 x+3)^{5/2} (13820 x+27937)}{6400}+\frac{321709971 \sqrt{1-2 x} \sqrt{5 x+3}}{204800}-\frac{3538809681 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{204800 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(321709971*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/204800 + (9748787*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/51200 + (33*Sqrt[1 -
2*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/20 + ((2 + 3*x)^3*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (9*Sqrt[1 - 2*x]*(3 + 5*x
)^(5/2)*(27937 + 13820*x))/6400 - (3538809681*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(204800*Sqrt[10])

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 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 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 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)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^2 (3+5 x)^{3/2} \left (52+\frac{165 x}{2}\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{33}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{1}{50} \int \frac{\left (-\frac{16505}{2}-\frac{51825 x}{4}\right ) (2+3 x) (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=\frac{33}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac{9748787 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{12800}\\ &=\frac{9748787 \sqrt{1-2 x} (3+5 x)^{3/2}}{51200}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac{321709971 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{102400}\\ &=\frac{321709971 \sqrt{1-2 x} \sqrt{3+5 x}}{204800}+\frac{9748787 \sqrt{1-2 x} (3+5 x)^{3/2}}{51200}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac{3538809681 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{409600}\\ &=\frac{321709971 \sqrt{1-2 x} \sqrt{3+5 x}}{204800}+\frac{9748787 \sqrt{1-2 x} (3+5 x)^{3/2}}{51200}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac{3538809681 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{204800 \sqrt{5}}\\ &=\frac{321709971 \sqrt{1-2 x} \sqrt{3+5 x}}{204800}+\frac{9748787 \sqrt{1-2 x} (3+5 x)^{3/2}}{51200}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac{3538809681 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{204800 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0468489, size = 79, normalized size = 0.51 \[ \frac{3538809681 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (13824000 x^5+65836800 x^4+148751040 x^3+233394520 x^2+381820658 x-538018839\right )}{2048000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-538018839 + 381820658*x + 233394520*x^2 + 148751040*x^3 + 65836800*x^4 + 13824000*x^5) +
3538809681*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(2048000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.013, size = 157, normalized size = 1. \begin{align*} -{\frac{1}{8192000\,x-4096000} \left ( -276480000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-1316736000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-2975020800\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-4667890400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-3538809681\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -7636413160\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4096000*(-276480000*x^5*(-10*x^2-x+3)^(1/2)-1316736000*x^4*(-10*x^2-x+3)^(1/2)-2975020800*x^3*(-10*x^2-x+3)
^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-4667890400*x^2*(-10*x^2-x+3)^(1/2)-3538809681*10^(1/2)*arcsi
n(20/11*x+1/11)-7636413160*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/
(2*x-1)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.86088, size = 170, normalized size = 1.1 \begin{align*} -\frac{675 \, x^{6}}{2 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{57915 \, x^{5}}{32 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{588291 \, x^{4}}{128 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{40330643 \, x^{3}}{5120 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{52185737 \, x^{2}}{4096 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3538809681}{4096000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1544632221 \, x}{204800 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1614056517}{204800 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-675/2*x^6/sqrt(-10*x^2 - x + 3) - 57915/32*x^5/sqrt(-10*x^2 - x + 3) - 588291/128*x^4/sqrt(-10*x^2 - x + 3) -
40330643/5120*x^3/sqrt(-10*x^2 - x + 3) - 52185737/4096*x^2/sqrt(-10*x^2 - x + 3) + 3538809681/4096000*sqrt(1
0)*arcsin(-20/11*x - 1/11) + 1544632221/204800*x/sqrt(-10*x^2 - x + 3) + 1614056517/204800/sqrt(-10*x^2 - x +
3)

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Fricas [A]  time = 1.90008, size = 343, normalized size = 2.23 \begin{align*} \frac{3538809681 \, \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 (13824000 \, x^{5} + 65836800 \, x^{4} + 148751040 \, x^{3} + 233394520 \, x^{2} + 381820658 \, x - 538018839\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{4096000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4096000*(3538809681*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*(13824000*x^5 + 65836800*x^4 + 148751040*x^3 + 233394520*x^2 + 381820658*x - 538018839)*sqrt(5*
x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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

[Out]

Timed out

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Giac [A]  time = 2.43018, size = 149, normalized size = 0.97 \begin{align*} -\frac{3538809681}{2048000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (4 \,{\left (24 \,{\left (36 \,{\left (16 \, \sqrt{5}{\left (5 \, x + 3\right )} + 141 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 42197 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 9748787 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 536183285 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 17694048405 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{25600000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3538809681/2048000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/25600000*(2*(4*(24*(36*(16*sqrt(5)*(5*x +
3) + 141*sqrt(5))*(5*x + 3) + 42197*sqrt(5))*(5*x + 3) + 9748787*sqrt(5))*(5*x + 3) + 536183285*sqrt(5))*(5*x
+ 3) - 17694048405*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)