3.2619 \(\int \frac{(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{1099 (3 x+2)^2}{726 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{\sqrt{1-2 x} (8200665 x+4898747)}{798600 \sqrt{5 x+3}}+\frac{4887 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

[Out]

(-1099*(2 + 3*x)^2)/(726*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (
Sqrt[1 - 2*x]*(4898747 + 8200665*x))/(798600*Sqrt[3 + 5*x]) + (4887*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqr
t[10])

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Rubi [A]  time = 0.0290192, antiderivative size = 113, normalized size of antiderivative = 1., 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} \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{1099 (3 x+2)^2}{726 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{\sqrt{1-2 x} (8200665 x+4898747)}{798600 \sqrt{5 x+3}}+\frac{4887 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1099*(2 + 3*x)^2)/(726*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (
Sqrt[1 - 2*x]*(4898747 + 8200665*x))/(798600*Sqrt[3 + 5*x]) + (4887*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqr
t[10])

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 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 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 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}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{33} \int \frac{(2+3 x)^2 \left (148+\frac{507 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac{1099 (2+3 x)^2}{726 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{363} \int \frac{\left (-\frac{14369}{2}-\frac{49701 x}{4}\right ) (2+3 x)}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{1099 (2+3 x)^2}{726 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{\sqrt{1-2 x} (4898747+8200665 x)}{798600 \sqrt{3+5 x}}+\frac{4887}{400} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{1099 (2+3 x)^2}{726 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{\sqrt{1-2 x} (4898747+8200665 x)}{798600 \sqrt{3+5 x}}+\frac{4887 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{200 \sqrt{5}}\\ &=-\frac{1099 (2+3 x)^2}{726 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{\sqrt{1-2 x} (4898747+8200665 x)}{798600 \sqrt{3+5 x}}+\frac{4887 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{200 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.109311, size = 65, normalized size = 0.58 \[ \frac{-6468660 x^3+40488772 x^2+12657123 x-8379147}{798600 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{4887 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8379147 + 12657123*x + 40488772*x^2 - 6468660*x^3)/(798600*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (4887*ArcSin[Sqr
t[5/11]*Sqrt[1 - 2*x]])/(200*Sqrt[10])

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Maple [A]  time = 0.013, size = 151, normalized size = 1.3 \begin{align*}{\frac{1}{15972000\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 390275820\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-156110328\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-129373200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-136596537\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+809775440\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-167582940\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15972000*(1-2*x)^(1/2)*(390275820*10^(1/2)*arcsin(20/11*x+1/11)*x^3-156110328*10^(1/2)*arcsin(20/11*x+1/11)*
x^2-129373200*x^3*(-10*x^2-x+3)^(1/2)-136596537*10^(1/2)*arcsin(20/11*x+1/11)*x+809775440*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)-167582940*(-10*x^2-x+3)^(1/2))/(2*x
-1)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 2.89055, size = 128, normalized size = 1.13 \begin{align*} \frac{4887}{4000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{81 \, x^{2}}{20 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{18627221 \, x}{798600 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3910543}{199650 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2401}{264 \,{\left (2 \, \sqrt{-10 \, x^{2} - x + 3} x - \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4887/4000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 81/20*x^2/sqrt(-10*x^2 - x + 3) - 18627221/798600*x/sqrt(-1
0*x^2 - x + 3) - 3910543/199650/sqrt(-10*x^2 - x + 3) - 2401/264/(2*sqrt(-10*x^2 - x + 3)*x - sqrt(-10*x^2 - x
+ 3))

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Fricas [A]  time = 1.53269, size = 340, normalized size = 3.01 \begin{align*} -\frac{19513791 \, \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 (6468660 \, x^{3} - 40488772 \, x^{2} - 12657123 \, x + 8379147\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{15972000 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15972000*(19513791*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*(6468660*x^3 - 40488772*x^2 - 12657123*x + 8379147)*sqrt(5*x + 3)*sqrt(-2*x +
1))/(20*x^3 - 8*x^2 - 7*x + 3)

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

[Out]

Timed out

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Giac [A]  time = 1.57236, size = 177, normalized size = 1.57 \begin{align*} \frac{4887}{2000} \, \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 )}}{332750 \, \sqrt{5 \, x + 3}} - \frac{{\left (4 \,{\left (323433 \, \sqrt{5}{\left (5 \, x + 3\right )} - 13033138 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 214579893 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{99825000 \,{\left (2 \, x - 1\right )}^{2}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{166375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4887/2000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/332750*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
)/sqrt(5*x + 3) - 1/99825000*(4*(323433*sqrt(5)*(5*x + 3) - 13033138*sqrt(5))*(5*x + 3) + 214579893*sqrt(5))*s
qrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 2/166375*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
)