3.2539 \(\int \frac{(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{13}{8} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^3+\frac{999}{160} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac{295101237 \sqrt{1-2 x} (5 x+3)^{3/2}}{409600}+\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)}{51200}+\frac{9738340821 \sqrt{1-2 x} \sqrt{5 x+3}}{1638400}-\frac{107121749031 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1638400 \sqrt{10}} \]

[Out]

(9738340821*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1638400 + (295101237*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/409600 + (999*Sqr
t[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/160 + (13*Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(5/2))/8 + ((2 + 3*x)^4*
(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(7611023 + 3765060*x))/51200 - (107121749031*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1638400*Sqrt[10])

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Rubi [A]  time = 0.0550919, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{13}{8} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^3+\frac{999}{160} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac{295101237 \sqrt{1-2 x} (5 x+3)^{3/2}}{409600}+\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)}{51200}+\frac{9738340821 \sqrt{1-2 x} \sqrt{5 x+3}}{1638400}-\frac{107121749031 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1638400 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9738340821*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1638400 + (295101237*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/409600 + (999*Sqr
t[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/160 + (13*Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(5/2))/8 + ((2 + 3*x)^4*
(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(7611023 + 3765060*x))/51200 - (107121749031*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1638400*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)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^3 (3+5 x)^{3/2} \left (61+\frac{195 x}{2}\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{1}{60} \int \frac{\left (-11805-\frac{74925 x}{4}\right ) (2+3 x)^2 (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{\int \frac{(2+3 x) (3+5 x)^{3/2} \left (\frac{7494225}{4}+\frac{23531625 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{3000}\\ &=\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{295101237 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{102400}\\ &=\frac{295101237 \sqrt{1-2 x} (3+5 x)^{3/2}}{409600}+\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{9738340821 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{819200}\\ &=\frac{9738340821 \sqrt{1-2 x} \sqrt{3+5 x}}{1638400}+\frac{295101237 \sqrt{1-2 x} (3+5 x)^{3/2}}{409600}+\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{107121749031 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3276800}\\ &=\frac{9738340821 \sqrt{1-2 x} \sqrt{3+5 x}}{1638400}+\frac{295101237 \sqrt{1-2 x} (3+5 x)^{3/2}}{409600}+\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{107121749031 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1638400 \sqrt{5}}\\ &=\frac{9738340821 \sqrt{1-2 x} \sqrt{3+5 x}}{1638400}+\frac{295101237 \sqrt{1-2 x} (3+5 x)^{3/2}}{409600}+\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{107121749031 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1638400 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.055032, size = 84, normalized size = 0.46 \[ \frac{107121749031 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (276480000 x^6+1479168000 x^5+3687379200 x^4+5945485120 x^3+7755469800 x^2+11734056318 x-16267424049\right )}{16384000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-16267424049 + 11734056318*x + 7755469800*x^2 + 5945485120*x^3 + 3687379200*x^4 + 14791680
00*x^5 + 276480000*x^6) + 107121749031*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(16384000*Sqrt[1 - 2*
x])

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Maple [A]  time = 0.012, size = 174, normalized size = 1. \begin{align*} -{\frac{1}{65536000\,x-32768000} \left ( -5529600000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}-29583360000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-73747584000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-118909702400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-155109396000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-107121749031\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -234681126360\,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)^4*(3+5*x)^(5/2)/(1-2*x)^(3/2),x)

[Out]

-1/32768000*(-5529600000*(-10*x^2-x+3)^(1/2)*x^6-29583360000*x^5*(-10*x^2-x+3)^(1/2)-73747584000*x^4*(-10*x^2-
x+3)^(1/2)-118909702400*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-155109396000*x^2*
(-10*x^2-x+3)^(1/2)-107121749031*10^(1/2)*arcsin(20/11*x+1/11)-234681126360*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 = 3.8259, size = 193, normalized size = 1.05 \begin{align*} -\frac{3375 \, x^{7}}{4 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{80325 \, x^{6}}{16 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3574125 \, x^{5}}{256 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{25493477 \, x^{4}}{1024 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1415345109 \, x^{3}}{40960 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{8193669099 \, x^{2}}{163840 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{107121749031}{32768000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{46134951291 \, x}{1638400 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{48802272147}{1638400 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/4*x^7/sqrt(-10*x^2 - x + 3) - 80325/16*x^6/sqrt(-10*x^2 - x + 3) - 3574125/256*x^5/sqrt(-10*x^2 - x + 3)
- 25493477/1024*x^4/sqrt(-10*x^2 - x + 3) - 1415345109/40960*x^3/sqrt(-10*x^2 - x + 3) - 8193669099/163840*x^
2/sqrt(-10*x^2 - x + 3) + 107121749031/32768000*sqrt(10)*arcsin(-20/11*x - 1/11) + 46134951291/1638400*x/sqrt(
-10*x^2 - x + 3) + 48802272147/1638400/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.77867, size = 382, normalized size = 2.09 \begin{align*} \frac{107121749031 \, \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 (276480000 \, x^{6} + 1479168000 \, x^{5} + 3687379200 \, x^{4} + 5945485120 \, x^{3} + 7755469800 \, x^{2} + 11734056318 \, x - 16267424049\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{32768000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32768000*(107121749031*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*(276480000*x^6 + 1479168000*x^5 + 3687379200*x^4 + 5945485120*x^3 + 7755469800*x^2 + 1173405
6318*x - 16267424049)*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)**4*(3+5*x)**(5/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 2.32109, size = 166, normalized size = 0.91 \begin{align*} -\frac{107121749031}{16384000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (4 \,{\left (8 \,{\left (108 \,{\left (16 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 35 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4299 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 3832457 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 295101237 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 16230568035 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 535608745155 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{204800000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-107121749031/16384000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/204800000*(2*(4*(8*(108*(16*(4*sqrt(5)
*(5*x + 3) + 35*sqrt(5))*(5*x + 3) + 4299*sqrt(5))*(5*x + 3) + 3832457*sqrt(5))*(5*x + 3) + 295101237*sqrt(5))
*(5*x + 3) + 16230568035*sqrt(5))*(5*x + 3) - 535608745155*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)