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]

104040277/1024000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-859837/76800*(3+5*x)^(3/2)*(1-2*x)^(1/2)-78167/
48000*(3+5*x)^(5/2)*(1-2*x)^(1/2)-963/4000*(3+5*x)^(7/2)*(1-2*x)^(1/2)-3/50*(2+3*x)*(3+5*x)^(7/2)*(1-2*x)^(1/2
)-9458207/102400*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 143, normalized size of antiderivative = 1.00, 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 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 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 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 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*}

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Mathematica [A]  time = 0.08, size = 88, normalized size = 0.62 \[ -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (6912000 x^4+26294400 x^3+44906720 x^2+48658820 x+46187289\right )[phone] \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{3072000 \sqrt {2 x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3072000*(Sqrt[1 - 2*x]*(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(46187289 + 48658820*x + 44906720*x^2 + 26294400*x^
3 + 6912000*x^4) + 312120831*Sqrt[10]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]]))/Sqrt[-1 + 2*x]

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fricas [A]  time = 1.22, size = 77, normalized size = 0.54 \[ -\frac {1}{307200} \, {\left (6912000 \, x^{4} + 26294400 \, x^{3} + 44906720 \, x^{2} + 48658820 \, x + 46187289\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {104040277}{2048000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

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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giac [A]  time = 1.10, size = 72, normalized size = 0.50 \[ -\frac {1}{15360000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (36 \, {\left (240 \, x + 481\right )} {\left (5 \, x + 3\right )} + 78167\right )} {\left (5 \, x + 3\right )} + 4299185\right )} {\left (5 \, x + 3\right )} + 141873105\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 1560604155 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]

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)))

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maple [A]  time = 0.01, size = 121, normalized size = 0.85 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-138240000 \sqrt {-10 x^{2}-x +3}\, x^{4}-525888000 \sqrt {-10 x^{2}-x +3}\, x^{3}-898134400 \sqrt {-10 x^{2}-x +3}\, x^{2}-973176400 \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-923745780 \sqrt {-10 x^{2}-x +3}\right )}{6144000 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6144000*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(-138240000*(-10*x^2-x+3)^(1/2)*x^4-525888000*(-10*x^2-x+3)^(1/2)*x^3-8
98134400*(-10*x^2-x+3)^(1/2)*x^2+312120831*10^(1/2)*arcsin(20/11*x+1/11)-973176400*(-10*x^2-x+3)^(1/2)*x-92374
5780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.25, size = 92, normalized size = 0.64 \[ -\frac {45}{2} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {2739}{32} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {280667}{1920} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {2432941}{15360} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {104040277}{2048000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {15395763}{102400} \, \sqrt {-10 \, x^{2} - x + 3} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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