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]

-107121749031/16384000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+(2+3*x)^4*(3+5*x)^(5/2)/(1-2*x)^(1/2)+2951
01237/409600*(3+5*x)^(3/2)*(1-2*x)^(1/2)+999/160*(2+3*x)^2*(3+5*x)^(5/2)*(1-2*x)^(1/2)+13/8*(2+3*x)^3*(3+5*x)^
(5/2)*(1-2*x)^(1/2)+1/51200*(3+5*x)^(5/2)*(7611023+3765060*x)*(1-2*x)^(1/2)[phone]/1638400*(1-2*x)^(1/2)*(
3+5*x)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 183, normalized size of antiderivative = 1.00, 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 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)^{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*}

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Mathematica [A]  time = 0.09, size = 98, normalized size = 0.54 \[ \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (276480000 x^6+1479168000 x^5+3687379200 x^4+5945485120 x^3+7755469800 x^2+11734056318 x-16267424049\right )-107121749031 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{16384000 \sqrt {-(1-2 x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-16267424049 + 11734056318*x + 7755469800*x^2 + 5945485120*x^3 + 3687379200
*x^4 + 1479168000*x^5 + 276480000*x^6) - 107121749031*Sqrt[10]*(-1 + 2*x)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/
(16384000*Sqrt[-(1 - 2*x)^2])

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fricas [A]  time = 0.89, size = 101, normalized size = 0.55 \[ \frac {107121749031 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (276480000 \, x^{6} + 1479168000 \, x^{5} + 3687379200 \, x^{4} + 5945485120 \, x^{3} + 7755469800 \, x^{2} + 11734056318 \, x - 16267424049\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{32768000 \, {\left (2 \, x - 1\right )}} \]

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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giac [A]  time = 1.22, size = 123, normalized size = 0.67 \[ -\frac {107121749031}{16384000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (8 \, {\left (108 \, {\left (16 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 35 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4299 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 3832457 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 295101237 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 16230568035 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 535608745155 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{204800000 \, {\left (2 \, x - 1\right )}} \]

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)

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maple [A]  time = 0.02, size = 174, normalized size = 0.95 \[ -\frac {\left (-5529600000 \sqrt {-10 x^{2}-x +3}\, x^{6}-29583360000 \sqrt {-10 x^{2}-x +3}\, x^{5}-73747584000 \sqrt {-10 x^{2}-x +3}\, x^{4}-118909702400 \sqrt {-10 x^{2}-x +3}\, x^{3}-155109396000 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-234681126360 \sqrt {-10 x^{2}-x +3}\, x -107121749031 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{32768000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32768000*(-5529600000*(-10*x^2-x+3)^(1/2)*x^6-29583360000*(-10*x^2-x+3)^(1/2)*x^5-73747584000*(-10*x^2-x+3)
^(1/2)*x^4-118909702400*(-10*x^2-x+3)^(1/2)*x^3+214243498062*10^(1/2)*x*arcsin(20/11*x+1/11)-155109396000*(-10
*x^2-x+3)^(1/2)*x^2-107121749031*10^(1/2)*arcsin(20/11*x+1/11)-234681126360*(-10*x^2-x+3)^(1/2)*x+325348480980
*(-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 [A]  time = 1.48, size = 143, normalized size = 0.78 \[ -\frac {3375 \, x^{7}}{4 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {80325 \, x^{6}}{16 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3574125 \, x^{5}}{256 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {25493477 \, x^{4}}{1024 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1415345109 \, x^{3}}{40960 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {8193669099 \, x^{2}}{163840 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {107121749031}{32768000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {46134951291 \, x}{1638400 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {48802272147}{1638400 \, \sqrt {-10 \, x^{2} - x + 3}} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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