3.26.72 \(\int \frac {(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\) [2572]

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

[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 = 0.02, 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 = {100, 155, 148, 56, 222} \begin {gather*} -\frac {621 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{100 \sqrt {10}}+\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}} \end {gather*}

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 56

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 100

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 148

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[m, 1] ||  !SumSimplerQ[n, 1])

Rule 155

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 222

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 \text {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 = 0.18, size = 69, normalized size = 0.61 \begin {gather*} \frac {3821563+11581424 x+6746215 x^2-3234330 x^3}{399300 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {621 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{100 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3821563 + 11581424*x + 6746215*x^2 - 3234330*x^3)/(399300*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (621*ArcTan[Sqrt[5
/2 - 5*x]/Sqrt[3 + 5*x]])/(100*Sqrt[10])

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Maple [A]
time = 0.08, size = 151, normalized size = 1.34

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \left (123982650 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-64686600 x^{3} \sqrt {-10 x^{2}-x +3}-29755836 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}-22316877 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{7986000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7986000*(1-2*x)^(1/2)*(123982650*10^(1/2)*arcsin(20/11*x+1/11)*x^3+86787855*10^(1/2)*arcsin(20/11*x+1/11)*x
^2-64686600*x^3*(-10*x^2-x+3)^(1/2)-29755836*10^(1/2)*arcsin(20/11*x+1/11)*x+134924300*x^2*(-10*x^2-x+3)^(1/2)
-22316877*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-1+2*x)
/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]
time = 0.50, size = 95, normalized size = 0.84 \begin {gather*} -\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 )}} \end {gather*}

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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Fricas [A]
time = 0.73, size = 106, normalized size = 0.94 \begin {gather*} \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 )}} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{4}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \end {gather*}

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]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (86) = 172\).
time = 1.39, size = 178, normalized size = 1.58 \begin {gather*} -\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}} \end {gather*}

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

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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