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

Optimal. Leaf size=157 \[ -\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{5 \sqrt {5 x+3}}+\frac {33}{125} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^2-\frac {9 (2127-460 x) (1-2 x)^{5/2} \sqrt {5 x+3}}{200000}+\frac {66997 (1-2 x)^{3/2} \sqrt {5 x+3}}{800000}+\frac {2210901 \sqrt {1-2 x} \sqrt {5 x+3}}{8000000}+\frac {24319911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000000 \sqrt {10}} \]

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Rubi [A]  time = 0.05, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 153, 147, 50, 54, 216} \begin {gather*} -\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{5 \sqrt {5 x+3}}+\frac {33}{125} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^2-\frac {9 (2127-460 x) (1-2 x)^{5/2} \sqrt {5 x+3}}{200000}+\frac {66997 (1-2 x)^{3/2} \sqrt {5 x+3}}{800000}+\frac {2210901 \sqrt {1-2 x} \sqrt {5 x+3}}{8000000}+\frac {24319911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) + (2210901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8000000 + (66997*(1
- 2*x)^(3/2)*Sqrt[3 + 5*x])/800000 - (9*(2127 - 460*x)*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/200000 + (33*(1 - 2*x)^
(5/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/125 + (24319911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8000000*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 97

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

Rule 147

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 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

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 {(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {(-1-33 x) (1-2 x)^{3/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{125} \int \frac {(1-2 x)^{3/2} (2+3 x) \left (-131+\frac {69 x}{2}\right )}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt {3+5 x}}-\frac {9 (2127-460 x) (1-2 x)^{5/2} \sqrt {3+5 x}}{200000}+\frac {33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {66997 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{80000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {66997 (1-2 x)^{3/2} \sqrt {3+5 x}}{800000}-\frac {9 (2127-460 x) (1-2 x)^{5/2} \sqrt {3+5 x}}{200000}+\frac {33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {2210901 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1600000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {2210901 \sqrt {1-2 x} \sqrt {3+5 x}}{8000000}+\frac {66997 (1-2 x)^{3/2} \sqrt {3+5 x}}{800000}-\frac {9 (2127-460 x) (1-2 x)^{5/2} \sqrt {3+5 x}}{200000}+\frac {33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {24319911 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16000000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {2210901 \sqrt {1-2 x} \sqrt {3+5 x}}{8000000}+\frac {66997 (1-2 x)^{3/2} \sqrt {3+5 x}}{800000}-\frac {9 (2127-460 x) (1-2 x)^{5/2} \sqrt {3+5 x}}{200000}+\frac {33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {24319911 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8000000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {2210901 \sqrt {1-2 x} \sqrt {3+5 x}}{8000000}+\frac {66997 (1-2 x)^{3/2} \sqrt {3+5 x}}{800000}-\frac {9 (2127-460 x) (1-2 x)^{5/2} \sqrt {3+5 x}}{200000}+\frac {33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {24319911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 93, normalized size = 0.59 \begin {gather*} \frac {24319911 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \left (69120000 x^6-9504000 x^5-91502400 x^4+31284920 x^3+44775890 x^2-8158469 x-6089453\right )}{80000000 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*(-6089453 - 8158469*x + 44775890*x^2 + 31284920*x^3 - 91502400*x^4 - 9504000*x^5 + 69120000*x^6) + 243199
11*Sqrt[3 + 5*x]*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(80000000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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IntegrateAlgebraic [A]  time = 0.28, size = 157, normalized size = 1.00 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {3200000 (1-2 x)^5}{(5 x+3)^5}+\frac {72499375 (1-2 x)^4}{(5 x+3)^4}+\frac {13945500 (1-2 x)^3}{(5 x+3)^3}-\frac {171512320 (1-2 x)^2}{(5 x+3)^2}-\frac {37518320 (1-2 x)}{5 x+3}-3215856\right )}{8000000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}-\frac {24319911 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{8000000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-121*Sqrt[1 - 2*x]*(-3215856 + (3200000*(1 - 2*x)^5)/(3 + 5*x)^5 + (72499375*(1 - 2*x)^4)/(3 + 5*x)^4 + (1394
5500*(1 - 2*x)^3)/(3 + 5*x)^3 - (171512320*(1 - 2*x)^2)/(3 + 5*x)^2 - (37518320*(1 - 2*x))/(3 + 5*x)))/(800000
0*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^5) - (24319911*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/
(8000000*Sqrt[10])

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fricas [A]  time = 1.23, size = 96, normalized size = 0.61 \begin {gather*} -\frac {24319911 \, \sqrt {10} {\left (5 \, 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 (34560000 \, x^{5} + 12528000 \, x^{4} - 39487200 \, x^{3} - 4101140 \, x^{2} + 20337375 \, x + 6089453\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{160000000 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/160000000*(24319911*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2
+ x - 3)) - 20*(34560000*x^5 + 12528000*x^4 - 39487200*x^3 - 4101140*x^2 + 20337375*x + 6089453)*sqrt(5*x + 3
)*sqrt(-2*x + 1))/(5*x + 3)

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giac [A]  time = 2.17, size = 150, normalized size = 0.96 \begin {gather*} \frac {1}{200000000} \, {\left (4 \, {\left (24 \, {\left (36 \, {\left (16 \, \sqrt {5} {\left (5 \, x + 3\right )} - 211 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 22859 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 969335 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 5816745 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {24319911}{80000000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {121 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{156250 \, \sqrt {5 \, x + 3}} + \frac {242 \, \sqrt {10} \sqrt {5 \, x + 3}}{78125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/200000000*(4*(24*(36*(16*sqrt(5)*(5*x + 3) - 211*sqrt(5))*(5*x + 3) + 22859*sqrt(5))*(5*x + 3) + 969335*sqrt
(5))*(5*x + 3) - 5816745*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 24319911/80000000*sqrt(10)*arcsin(1/11*sqrt(
22)*sqrt(5*x + 3)) - 121/156250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 242/78125*sqrt(1
0)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

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maple [A]  time = 0.02, size = 150, normalized size = 0.96 \begin {gather*} \frac {\left (691200000 \sqrt {-10 x^{2}-x +3}\, x^{5}[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}-789744000 \sqrt {-10 x^{2}-x +3}\, x^{3}-82022800 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{160000000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/160000000*(691200000*(-10*x^2-x+3)^(1/2)*x^5+250560000*(-10*x^2-x+3)^(1/2)*x^4-789744000*(-10*x^2-x+3)^(1/2)
*x^3+121599555*10^(1/2)*x*arcsin(20/11*x+1/11)-82022800*(-10*x^2-x+3)^(1/2)*x^2+72959733*10^(1/2)*arcsin(20/11
*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x+121789060*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(-10*x^2-x+3)^(1/2)/(5*
x+3)^(1/2)

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maxima [A]  time = 1.19, size = 126, normalized size = 0.80 \begin {gather*} -\frac {216 \, x^{6}}{25 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {297 \, x^{5}}{250 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {57189 \, x^{4}}{5000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {782123 \, x^{3}}{200000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {4477589 \, x^{2}}{800000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {24319911}{160000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {8158469 \, x}{8000000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {6089453}{8000000 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-216/25*x^6/sqrt(-10*x^2 - x + 3) + 297/250*x^5/sqrt(-10*x^2 - x + 3) + 57189/5000*x^4/sqrt(-10*x^2 - x + 3) -
782123/200000*x^3/sqrt(-10*x^2 - x + 3) - 4477589/800000*x^2/sqrt(-10*x^2 - x + 3) - 24319911/160000000*sqrt(
10)*arcsin(-20/11*x - 1/11) + 8158469/8000000*x/sqrt(-10*x^2 - x + 3) + 6089453/8000000/sqrt(-10*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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