3.27.18 \(\int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\) [2618]

Optimal. Leaf size=142 \[ \frac {7723 \sqrt {1-2 x} (2+3 x)^2}{39930 \sqrt {3+5 x}}-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (39109961+16227780 x)}{2129600}+\frac {243189 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}} \]

[Out]

243189/16000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+7/33*(2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^(1/2)-1561/726*
(2+3*x)^3/(1-2*x)^(1/2)/(3+5*x)^(1/2)+7723/39930*(2+3*x)^2*(1-2*x)^(1/2)/(3+5*x)^(1/2)-1/2129600*(39109961+162
27780*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 155, 152, 56, 222} \begin {gather*} \frac {243189 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}}+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {1561 (3 x+2)^3}{726 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {7723 \sqrt {1-2 x} (3 x+2)^2}{39930 \sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (16227780 x+39109961)}{2129600} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7723*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*Sqrt[3 + 5*x]) - (1561*(2 + 3*x)^3)/(726*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
+ (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(39109961 + 16227780*x))/2
129600 + (243189*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*Sqrt[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 152

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 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)^5}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {(2+3 x)^3 \left (211+\frac {717 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{363} \int \frac {\left (-17212-\frac {117321 x}{4}\right ) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {7723 \sqrt {1-2 x} (2+3 x)^2}{39930 \sqrt {3+5 x}}-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {2 \int \frac {\left (-\frac {1244193}{4}-\frac {4056945 x}{8}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{19965}\\ &=\frac {7723 \sqrt {1-2 x} (2+3 x)^2}{39930 \sqrt {3+5 x}}-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (39109961+16227780 x)}{2129600}+\frac {243189 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3200}\\ &=\frac {7723 \sqrt {1-2 x} (2+3 x)^2}{39930 \sqrt {3+5 x}}-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (39109961+16227780 x)}{2129600}+\frac {243189 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1600 \sqrt {5}}\\ &=\frac {7723 \sqrt {1-2 x} (2+3 x)^2}{39930 \sqrt {3+5 x}}-\frac {1561 (2+3 x)^3}{726 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (39109961+16227780 x)}{2129600}+\frac {243189 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 93, normalized size = 0.65 \begin {gather*} -\frac {10 \sqrt {3+5 x} \left (435258129-525679641 x-1790987404 x^2+536898780 x^3+77623920 x^4\right )-971053677 \sqrt {10-20 x} \left (-3+x+10 x^2\right ) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{63888000 (1-2 x)^{3/2} (3+5 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/63888000*(10*Sqrt[3 + 5*x]*(435258129 - 525679641*x - 1790987404*x^2 + 536898780*x^3 + 77623920*x^4) - 9710
53677*Sqrt[10 - 20*x]*(-3 + x + 10*x^2)*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/((1 - 2*x)^(3/2)*(3 + 5*x))

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Maple [A]
time = 0.09, size = 168, normalized size = 1.18

method	result	size
default	\(\frac {\sqrt {1-2 x}\, \left (19421073540 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-1552478400 x^{4} \sqrt {-10 x^{2}-x +3}-7768429416 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-10737975600 x^{3} \sqrt {-10 x^{2}-x +3}-6797375739 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-8705162580 \sqrt {-10 x^{2}-x +3}\right )}{127776000 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/127776000*(1-2*x)^(1/2)*(19421073540*10^(1/2)*arcsin(20/11*x+1/11)*x^3-1552478400*x^4*(-10*x^2-x+3)^(1/2)-77
68429416*10^(1/2)*arcsin(20/11*x+1/11)*x^2-10737975600*x^3*(-10*x^2-x+3)^(1/2)-6797375739*10^(1/2)*arcsin(20/1
1*x+1/11)*x+35819748080*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^
2-x+3)^(1/2)-8705162580*(-10*x^2-x+3)^(1/2))/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]
time = 0.51, size = 112, normalized size = 0.79 \begin {gather*} \frac {243 \, x^{3}}{40 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {243189}{32000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {7209 \, x^{2}}{160 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {751566017 \, x}{6388800 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {638622829}{6388800 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {16807}{528 \, {\left (2 \, \sqrt {-10 \, x^{2} - x + 3} x - \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/40*x^3/sqrt(-10*x^2 - x + 3) + 243189/32000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 7209/160*x^2/sqrt(-10
*x^2 - x + 3) - 751566017/6388800*x/sqrt(-10*x^2 - x + 3) - 638622829/6388800/sqrt(-10*x^2 - x + 3) - 16807/52
8/(2*sqrt(-10*x^2 - x + 3)*x - sqrt(-10*x^2 - x + 3))

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Fricas [A]
time = 0.45, size = 111, normalized size = 0.78 \begin {gather*} -\frac {971053677 \, \sqrt {10} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 (77623920 \, x^{4} + 536898780 \, x^{3} - 1790987404 \, x^{2} - 525679641 \, x + 435258129\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{127776000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/127776000*(971053677*sqrt(10)*(20*x^3 - 8*x^2 - 7*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt
(-2*x + 1)/(10*x^2 + x - 3)) + 20*(77623920*x^4 + 536898780*x^3 - 1790987404*x^2 - 525679641*x + 435258129)*sq
rt(5*x + 3)*sqrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.38, size = 144, normalized size = 1.01 \begin {gather*} \frac {243189}{16000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1663750 \, \sqrt {5 \, x + 3}} - \frac {{\left (4 \, {\left (323433 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 271 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 3237172310 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 53407238379 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{3993000000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{831875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243189/16000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/1663750*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))/sqrt(5*x + 3) - 1/3993000000*(4*(323433*(12*sqrt(5)*(5*x + 3) + 271*sqrt(5))*(5*x + 3) - 3237172310*sqrt
(5))*(5*x + 3) + 53407238379*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 2/831875*sqrt(10)*sqrt(5*x +
3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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