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

Optimal. Leaf size=138 \[ -\frac {9}{200} \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {5 x+3}}+\frac {651 \sqrt {5 x+3} (1-2 x)^{5/2}}{22000}+\frac {651 \sqrt {5 x+3} (1-2 x)^{3/2}}{8000}+\frac {21483 \sqrt {5 x+3} \sqrt {1-2 x}}{80000}+\frac {236313 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80000 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 138, 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 = {89, 80, 50, 54, 216} \begin {gather*} -\frac {9}{200} \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {5 x+3}}+\frac {651 \sqrt {5 x+3} (1-2 x)^{5/2}}{22000}+\frac {651 \sqrt {5 x+3} (1-2 x)^{3/2}}{8000}+\frac {21483 \sqrt {5 x+3} \sqrt {1-2 x}}{80000}+\frac {236313 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(7/2))/(275*Sqrt[3 + 5*x]) + (21483*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/80000 + (651*(1 - 2*x)^(3/2)*Sq
rt[3 + 5*x])/8000 + (651*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/22000 - (9*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/200 + (23631
3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80000*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 89

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

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)^2}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {2}{275} \int \frac {(1-2 x)^{5/2} \left (\frac {351}{2}+\frac {495 x}{2}\right )}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {1953 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{4400}\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {651}{800} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {21483 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{16000}\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{160000}\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{80000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{80000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 88, normalized size = 0.64 \begin {gather*} \frac {236313 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \left (288000 x^5-299200 x^4-147640 x^3+381870 x^2+24773 x-79699\right )}{800000 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*(-79699 + 24773*x + 381870*x^2 - 147640*x^3 - 299200*x^4 + 288000*x^5) + 236313*Sqrt[3 + 5*x]*Sqrt[-10 +
20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(800000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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IntegrateAlgebraic [A]  time = 0.23, size = 141, normalized size = 1.02 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {32000 (1-2 x)^4}{(5 x+3)^4}-\frac {88675 (1-2 x)^3}{(5 x+3)^3}-\frac {475230 (1-2 x)^2}{(5 x+3)^2}-\frac {143220 (1-2 x)}{5 x+3}-15624\right )}{80000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}-\frac {236313 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{80000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-121*Sqrt[1 - 2*x]*(-15624 + (32000*(1 - 2*x)^4)/(3 + 5*x)^4 - (88675*(1 - 2*x)^3)/(3 + 5*x)^3 - (475230*(1 -
2*x)^2)/(3 + 5*x)^2 - (143220*(1 - 2*x))/(3 + 5*x)))/(80000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^4) -
(236313*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(80000*Sqrt[10])

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fricas [A]  time = 0.83, size = 91, normalized size = 0.66 \begin {gather*} -\frac {236313 \, \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 (144000 \, x^{4} - 77600 \, x^{3} - 112620 \, x^{2} + 134625 \, x + 79699\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1600000 \, {\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)^2/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-1/1600000*(236313*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*(144000*x^4 - 77600*x^3 - 112620*x^2 + 134625*x + 79699)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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giac [A]  time = 1.92, size = 137, normalized size = 0.99 \begin {gather*} \frac {1}{2000000} \, {\left (4 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 529 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 16905 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 61545 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {236313}{800000} \, \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 )}}{31250 \, \sqrt {5 \, x + 3}} + \frac {242 \, \sqrt {10} \sqrt {5 \, x + 3}}{15625 \, {\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)^2/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

1/2000000*(4*(8*(36*sqrt(5)*(5*x + 3) - 529*sqrt(5))*(5*x + 3) + 16905*sqrt(5))*(5*x + 3) + 61545*sqrt(5))*sqr
t(5*x + 3)*sqrt(-10*x + 5) + 236313/800000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 121/31250*sqrt(10)*(
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 242/15625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))

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maple [A]  time = 0.01, size = 133, normalized size = 0.96 \begin {gather*} \frac {\left (2880000 \sqrt {-10 x^{2}-x +3}\, x^{4}-1552000 \sqrt {-10 x^{2}-x +3}\, x^{3}-2252400 \sqrt {-10 x^{2}-x +3}\, x^{2}+1181565 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2692500 \sqrt {-10 x^{2}-x +3}\, x +708939 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1593980 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{1600000 \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)^2/(5*x+3)^(3/2),x)

[Out]

1/1600000*(2880000*(-10*x^2-x+3)^(1/2)*x^4-1552000*(-10*x^2-x+3)^(1/2)*x^3+1181565*10^(1/2)*x*arcsin(20/11*x+1
/11)-2252400*(-10*x^2-x+3)^(1/2)*x^2+708939*10^(1/2)*arcsin(20/11*x+1/11)+2692500*(-10*x^2-x+3)^(1/2)*x+159398
0*(-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 = 1.37, size = 109, normalized size = 0.79 \begin {gather*} -\frac {18 \, x^{5}}{5 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {187 \, x^{4}}{50 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3691 \, x^{3}}{2000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {38187 \, x^{2}}{8000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {236313}{1600000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {24773 \, x}{80000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {79699}{80000 \, \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)^2/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-18/5*x^5/sqrt(-10*x^2 - x + 3) + 187/50*x^4/sqrt(-10*x^2 - x + 3) + 3691/2000*x^3/sqrt(-10*x^2 - x + 3) - 381
87/8000*x^2/sqrt(-10*x^2 - x + 3) - 236313/1600000*sqrt(10)*arcsin(-20/11*x - 1/11) - 24773/80000*x/sqrt(-10*x
^2 - x + 3) + 79699/80000/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 )}^2}{{\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)^2)/(5*x + 3)^(3/2),x)

[Out]

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

[Out]

Timed out

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