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

Optimal. Leaf size=74 \[ -\frac {2 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {5 x+3}}+\frac {4}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {47, 54, 216} \begin {gather*} -\frac {2 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {5 x+3}}+\frac {4}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + (4*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*x]) + (4*Sqrt[2/5]*ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]])/25

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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 {(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {2}{5} \int \frac {\sqrt {1-2 x}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}+\frac {4}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}+\frac {8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac {4 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}+\frac {4}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 70, normalized size = 0.95 \begin {gather*} \frac {-800 x^2+140 x+12 (5 x+3)^{3/2} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )+130}{375 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(130 + 140*x - 800*x^2 + 12*(3 + 5*x)^(3/2)*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(375*Sqrt[1 -
2*x]*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.08, size = 77, normalized size = 1.04 \begin {gather*} -\frac {2 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}-6\right )}{75 \sqrt {5 x+3}}-\frac {4}{25} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(-6 + (5*(1 - 2*x))/(3 + 5*x)))/(75*Sqrt[3 + 5*x]) - (4*Sqrt[2/5]*ArcTan[(Sqrt[5/2]*Sqrt[1 -
2*x])/Sqrt[3 + 5*x]])/25

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fricas [A]  time = 1.63, size = 92, normalized size = 1.24 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {5} \sqrt {2} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 5 \, {\left (40 \, x + 13\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}\right )}}{375 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/375*(3*sqrt(5)*sqrt(2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 5*(40*x + 13)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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giac [B]  time = 1.30, size = 139, normalized size = 1.88 \begin {gather*} -\frac {1}{6000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {60 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {4}{125} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {15 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 60*(sqrt(2)*sqrt(-10*x + 5) - sqrt(
22))/sqrt(5*x + 3)) + 4/125*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/375*sqrt(10)*(5*x + 3)^(3/2)*(15*
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2 x +1\right )^{\frac {3}{2}}}{\left (5 x +3\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.56, size = 93, normalized size = 1.26 \begin {gather*} \frac {2}{125} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{15 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} - \frac {11 \, \sqrt {-10 \, x^{2} - x + 3}}{75 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {14 \, \sqrt {-10 \, x^{2} - x + 3}}{75 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/125*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/15*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x + 27) -
11/75*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) + 14/75*sqrt(-10*x^2 - x + 3)/(5*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 )}^{3/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 3.33, size = 204, normalized size = 2.76 \begin {gather*} \begin {cases} \frac {16 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{375} - \frac {22 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{1875 \left (x + \frac {3}{5}\right )} + \frac {2 \sqrt {10} i \log {\left (\frac {1}{x + \frac {3}{5}} \right )}}{125} + \frac {2 \sqrt {10} i \log {\left (x + \frac {3}{5} \right )}}{125} + \frac {4 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{125} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\\frac {16 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{375} - \frac {22 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{1875 \left (x + \frac {3}{5}\right )} + \frac {2 \sqrt {10} i \log {\left (\frac {1}{x + \frac {3}{5}} \right )}}{125} - \frac {4 \sqrt {10} i \log {\left (\sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} + 1 \right )}}{125} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/375 - 22*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/(1875*(x +
3/5)) + 2*sqrt(10)*I*log(1/(x + 3/5))/125 + 2*sqrt(10)*I*log(x + 3/5)/125 + 4*sqrt(10)*asin(sqrt(110)*sqrt(x +
3/5)/11)/125, 11/(10*Abs(x + 3/5)) > 1), (16*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/375 - 22*sqrt(10)*I*sqrt(
1 - 11/(10*(x + 3/5)))/(1875*(x + 3/5)) + 2*sqrt(10)*I*log(1/(x + 3/5))/125 - 4*sqrt(10)*I*log(sqrt(1 - 11/(10
*(x + 3/5))) + 1)/125, True))

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