3.2479 \(\int \frac{(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{3}{50} \sqrt{1-2 x} (3 x+2) (5 x+3)^{7/2}-\frac{963 \sqrt{1-2 x} (5 x+3)^{7/2}}{4000}-\frac{78167 \sqrt{1-2 x} (5 x+3)^{5/2}}{48000}-\frac{859837 \sqrt{1-2 x} (5 x+3)^{3/2}}{76800}-\frac{9458207 \sqrt{1-2 x} \sqrt{5 x+3}}{102400}+\frac{104040277 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]

[Out]

(-9458207*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400 - (859837*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/76800 - (78167*Sqrt[1 -
2*x]*(3 + 5*x)^(5/2))/48000 - (963*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/4000 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)
^(7/2))/50 + (104040277*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400*Sqrt[10])

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Rubi [A]  time = 0.0398347, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac{3}{50} \sqrt{1-2 x} (3 x+2) (5 x+3)^{7/2}-\frac{963 \sqrt{1-2 x} (5 x+3)^{7/2}}{4000}-\frac{78167 \sqrt{1-2 x} (5 x+3)^{5/2}}{48000}-\frac{859837 \sqrt{1-2 x} (5 x+3)^{3/2}}{76800}-\frac{9458207 \sqrt{1-2 x} \sqrt{5 x+3}}{102400}+\frac{104040277 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9458207*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400 - (859837*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/76800 - (78167*Sqrt[1 -
2*x]*(3 + 5*x)^(5/2))/48000 - (963*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/4000 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)
^(7/2))/50 + (104040277*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400*Sqrt[10])

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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)^2 (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx &=-\frac{3}{50} \sqrt{1-2 x} (2+3 x) (3+5 x)^{7/2}-\frac{1}{50} \int \frac{\left (-314-\frac{963 x}{2}\right ) (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{963 \sqrt{1-2 x} (3+5 x)^{7/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac{78167 \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx}{8000}\\ &=-\frac{78167 \sqrt{1-2 x} (3+5 x)^{5/2}}{48000}-\frac{963 \sqrt{1-2 x} (3+5 x)^{7/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac{859837 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{19200}\\ &=-\frac{859837 \sqrt{1-2 x} (3+5 x)^{3/2}}{76800}-\frac{78167 \sqrt{1-2 x} (3+5 x)^{5/2}}{48000}-\frac{963 \sqrt{1-2 x} (3+5 x)^{7/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac{9458207 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{51200}\\ &=-\frac{9458207 \sqrt{1-2 x} \sqrt{3+5 x}}{102400}-\frac{859837 \sqrt{1-2 x} (3+5 x)^{3/2}}{76800}-\frac{78167 \sqrt{1-2 x} (3+5 x)^{5/2}}{48000}-\frac{963 \sqrt{1-2 x} (3+5 x)^{7/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac{104040277 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{204800}\\ &=-\frac{9458207 \sqrt{1-2 x} \sqrt{3+5 x}}{102400}-\frac{859837 \sqrt{1-2 x} (3+5 x)^{3/2}}{76800}-\frac{78167 \sqrt{1-2 x} (3+5 x)^{5/2}}{48000}-\frac{963 \sqrt{1-2 x} (3+5 x)^{7/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac{104040277 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{102400 \sqrt{5}}\\ &=-\frac{9458207 \sqrt{1-2 x} \sqrt{3+5 x}}{102400}-\frac{859837 \sqrt{1-2 x} (3+5 x)^{3/2}}{76800}-\frac{78167 \sqrt{1-2 x} (3+5 x)^{5/2}}{48000}-\frac{963 \sqrt{1-2 x} (3+5 x)^{7/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac{104040277 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{102400 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0474077, size = 70, normalized size = 0.49 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4+26294400 x^3+44906720 x^2+48658820 x+46187289\right )-312120831 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3072000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(46187289 + 48658820*x + 44906720*x^2 + 26294400*x^3 + 6912000*x^4) - 3121208
31*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/3072000

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Maple [A]  time = 0.009, size = 121, normalized size = 0.9 \begin{align*}{\frac{1}{6144000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-525888000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-898134400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -973176400\,x\sqrt{-10\,{x}^{2}-x+3}-923745780\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6144000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-138240000*x^4*(-10*x^2-x+3)^(1/2)-525888000*x^3*(-10*x^2-x+3)^(1/2)-89
8134400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-973176400*x*(-10*x^2-x+3)^(1/2)-923745
780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.23567, size = 124, normalized size = 0.87 \begin{align*} -\frac{45}{2} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{2739}{32} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{280667}{1920} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{2432941}{15360} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{104040277}{2048000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{15395763}{102400} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/2*sqrt(-10*x^2 - x + 3)*x^4 - 2739/32*sqrt(-10*x^2 - x + 3)*x^3 - 280667/1920*sqrt(-10*x^2 - x + 3)*x^2 -
2432941/15360*sqrt(-10*x^2 - x + 3)*x - 104040277/2048000*sqrt(10)*arcsin(-20/11*x - 1/11) - 15395763/102400*s
qrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.82836, size = 292, normalized size = 2.04 \begin{align*} -\frac{1}{307200} \,{\left (6912000 \, x^{4} + 26294400 \, x^{3} + 44906720 \, x^{2} + 48658820 \, x + 46187289\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{104040277}{2048000} \, \sqrt{10} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/307200*(6912000*x^4 + 26294400*x^3 + 44906720*x^2 + 48658820*x + 46187289)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1
04040277/2048000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + 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)**2*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.55077, size = 97, normalized size = 0.68 \begin{align*} -\frac{1}{15360000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (36 \,{\left (240 \, x + 481\right )}{\left (5 \, x + 3\right )} + 78167\right )}{\left (5 \, x + 3\right )} + 4299185\right )}{\left (5 \, x + 3\right )} + 141873105\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 1560604155 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15360000*sqrt(5)*(2*(4*(8*(36*(240*x + 481)*(5*x + 3) + 78167)*(5*x + 3) + 4299185)*(5*x + 3) + 141873105)*
sqrt(5*x + 3)*sqrt(-10*x + 5) - 1560604155*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))