3.2572 \(\int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac {7 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {107 \sqrt {1-2 x} (3 x+2)^2}{1815 (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x} (1051875 x+627641)}{399300 \sqrt {5 x+3}}-\frac {621 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{100 \sqrt {10}} \]

[Out]

-621/1000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+7/11*(2+3*x)^3/(3+5*x)^(3/2)/(1-2*x)^(1/2)-107/1815*(2+
3*x)^2*(1-2*x)^(1/2)/(3+5*x)^(3/2)+1/399300*(627641+1051875*x)*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 150, 143, 54, 216} \[ \frac {7 (3 x+2)^3}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {107 \sqrt {1-2 x} (3 x+2)^2}{1815 (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x} (1051875 x+627641)}{399300 \sqrt {5 x+3}}-\frac {621 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{100 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^3)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) +
(Sqrt[1 - 2*x]*(627641 + 1051875*x))/(399300*Sqrt[3 + 5*x]) - (621*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(100*Sqr
t[10])

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 98

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

Rule 143

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

Rule 150

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

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}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1}{11} \int \frac {(2+3 x)^2 \left (82+\frac {309 x}{2}\right )}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^2}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2 \int \frac {(2+3 x) \left (\frac {9127}{2}+\frac {31875 x}{4}\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{1815}\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^2}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x} (627641+1051875 x)}{399300 \sqrt {3+5 x}}-\frac {621}{200} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^2}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x} (627641+1051875 x)}{399300 \sqrt {3+5 x}}-\frac {621 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{100 \sqrt {5}}\\ &=-\frac {107 \sqrt {1-2 x} (2+3 x)^2}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^3}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x} (627641+1051875 x)}{399300 \sqrt {3+5 x}}-\frac {621 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{100 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 90, normalized size = 0.80 \[ \frac {10 \sqrt {2 x-1} \left (-3234330 x^3+6746215 x^2+11581424 x+3821563\right )-2479653 \sqrt {50 x+30} \left (10 x^2+x-3\right ) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{3993000 \sqrt {-(1-2 x)^2} (5 x+3)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(10*Sqrt[-1 + 2*x]*(3821563 + 11581424*x + 6746215*x^2 - 3234330*x^3) - 2479653*Sqrt[30 + 50*x]*(-3 + x + 10*x
^2)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(3993000*Sqrt[-(1 - 2*x)^2]*(3 + 5*x)^(3/2))

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fricas [A]  time = 0.70, size = 106, normalized size = 0.94 \[ \frac {2479653 \, \sqrt {10} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (3234330 \, x^{3} - 6746215 \, x^{2} - 11581424 \, x - 3821563\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7986000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7986000*(2479653*sqrt(10)*(50*x^3 + 35*x^2 - 12*x - 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2
*x + 1)/(10*x^2 + x - 3)) + 20*(3234330*x^3 - 6746215*x^2 - 11581424*x - 3821563)*sqrt(5*x + 3)*sqrt(-2*x + 1)
)/(50*x^3 + 35*x^2 - 12*x - 9)

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giac [B]  time = 1.47, size = 178, normalized size = 1.58 \[ -\frac {1}{39930000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {3252 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {621}{1000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (215622 \, \sqrt {5} {\left (5 \, x + 3\right )} - 4187171 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{16637500 \, {\left (2 \, x - 1\right )}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {813 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{2495625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/39930000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 3252*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3)) - 621/1000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/16637500*(215622*sqrt(5)
*(5*x + 3) - 4187171*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1/2495625*sqrt(10)*(5*x + 3)^(3/2)*(81
3*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3

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maple [A]  time = 0.02, size = 151, normalized size = 1.34 \[ -\frac {\sqrt {-2 x +1}\, \left (123982650 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-64686600 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}-29755836 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x -22316877 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{7986000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7986000*(-2*x+1)^(1/2)*(123982650*10^(1/2)*x^3*arcsin(20/11*x+1/11)[phone]*10^(1/2)*x^2*arcsin(20/11*x+1/
11)-64686600*(-10*x^2-x+3)^(1/2)*x^3-29755836*10^(1/2)*x*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x^
2-22316877*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x+76431260*(-10*x^2-x+3)^(1/2))/(2*x-1)
/(-10*x^2-x+3)^(1/2)/(5*x+3)^(3/2)

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maxima [A]  time = 1.23, size = 95, normalized size = 0.84 \[ -\frac {621}{2000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {81 \, x^{2}}{50 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {8686813 \, x}{1996500 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {31846681}{9982500 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{20625 \, {\left (5 \, \sqrt {-10 \, x^{2} - x + 3} x + 3 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-621/2000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 81/50*x^2/sqrt(-10*x^2 - x + 3) + 8686813/1996500*x/sqrt(-1
0*x^2 - x + 3) + 31846681/9982500/sqrt(-10*x^2 - x + 3) - 2/20625/(5*sqrt(-10*x^2 - x + 3)*x + 3*sqrt(-10*x^2
- x + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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