3.2563 \(\int \frac{(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{7 (3 x+2)^2}{11 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{\sqrt{1-2 x} (50985 x+30443)}{12100 \sqrt{5 x+3}}-\frac{999 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (Sqrt[1 - 2*x]*(30443 + 50985*x))/(12100*Sqrt[3 + 5*x]) - (
999*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(100*Sqrt[10])

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Rubi [A]  time = 0.0210367, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {98, 143, 54, 216} \[ \frac{7 (3 x+2)^2}{11 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{\sqrt{1-2 x} (50985 x+30443)}{12100 \sqrt{5 x+3}}-\frac{999 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (Sqrt[1 - 2*x]*(30443 + 50985*x))/(12100*Sqrt[3 + 5*x]) - (
999*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(100*Sqrt[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 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)^3}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^2}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{1}{11} \int \frac{(2+3 x) \left (89+\frac{309 x}{2}\right )}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{7 (2+3 x)^2}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{\sqrt{1-2 x} (30443+50985 x)}{12100 \sqrt{3+5 x}}-\frac{999}{200} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{7 (2+3 x)^2}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{\sqrt{1-2 x} (30443+50985 x)}{12100 \sqrt{3+5 x}}-\frac{999 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{100 \sqrt{5}}\\ &=\frac{7 (2+3 x)^2}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{\sqrt{1-2 x} (30443+50985 x)}{12100 \sqrt{3+5 x}}-\frac{999 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{100 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0532527, size = 70, normalized size = 0.83 \[ \frac{-326700 x^2+824990 x+120879 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+612430}{121000 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(612430 + 824990*x - 326700*x^2 + 120879*Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1210
00*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.013, size = 120, normalized size = 1.4 \begin{align*} -{\frac{1}{484000\,x-242000}\sqrt{1-2\,x} \left ( 1208790\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+120879\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-653400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-362637\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1649980\,x\sqrt{-10\,{x}^{2}-x+3}+1224860\,\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)^3/(1-2*x)^(3/2)/(3+5*x)^(3/2),x)

[Out]

-1/242000*(1-2*x)^(1/2)*(1208790*10^(1/2)*arcsin(20/11*x+1/11)*x^2+120879*10^(1/2)*arcsin(20/11*x+1/11)*x-6534
00*x^2*(-10*x^2-x+3)^(1/2)-362637*10^(1/2)*arcsin(20/11*x+1/11)+1649980*x*(-10*x^2-x+3)^(1/2)+1224860*(-10*x^2
-x+3)^(1/2))/(2*x-1)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.75829, size = 78, normalized size = 0.93 \begin{align*} -\frac{27 \, x^{2}}{10 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{999}{2000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{82499 \, x}{12100 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{61243}{12100 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/10*x^2/sqrt(-10*x^2 - x + 3) + 999/2000*sqrt(10)*arcsin(-20/11*x - 1/11) + 82499/12100*x/sqrt(-10*x^2 - x
+ 3) + 61243/12100/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.75342, size = 277, normalized size = 3.3 \begin{align*} \frac{120879 \, \sqrt{10}{\left (10 \, x^{2} + 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 (32670 \, x^{2} - 82499 \, x - 61243\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{242000 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/242000*(120879*sqrt(10)*(10*x^2 + x - 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^
2 + x - 3)) + 20*(32670*x^2 - 82499*x - 61243)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(10*x^2 + x - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.28557, size = 159, normalized size = 1.89 \begin{align*} -\frac{999}{1000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6534 \, \sqrt{5}{\left (5 \, x + 3\right )} - 121687 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{302500 \,{\left (2 \, x - 1\right )}} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{30250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{15125 \,{\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)^3/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-999/1000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/302500*(6534*sqrt(5)*(5*x + 3) - 121687*sqrt(5))*sq
rt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 1/30250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) +
2/15125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))