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

Optimal. Leaf size=74 \[ -\frac {2 (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}-\frac {6}{25} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {33}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {47, 50, 54, 216} \begin {gather*} -\frac {2 (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}-\frac {6}{25} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {33}{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)^(3/2),x]

[Out]

(-2*(1 - 2*x)^(3/2))/(5*Sqrt[3 + 5*x]) - (6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25 - (33*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 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 {(1-2 x)^{3/2}}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{3/2}}{5 \sqrt {3+5 x}}-\frac {6}{5} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{5 \sqrt {3+5 x}}-\frac {6}{25} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {33}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{5 \sqrt {3+5 x}}-\frac {6}{25} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {66 \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}}{5 \sqrt {3+5 x}}-\frac {6}{25} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {33}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 39, normalized size = 0.53 \begin {gather*} -\frac {2}{55} \sqrt {\frac {2}{11}} (1-2 x)^{5/2} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-\frac {5}{11} (2 x-1)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2/11]*(1 - 2*x)^(5/2)*Hypergeometric2F1[3/2, 5/2, 7/2, (-5*(-1 + 2*x))/11])/55

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IntegrateAlgebraic [A]  time = 0.28, size = 96, normalized size = 1.30 \begin {gather*} \frac {66}{25} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )-\frac {2 \sqrt {11-2 (5 x+3)} \left (\sqrt {5} (5 x+3)+11 \sqrt {5}\right )}{125 \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[11 - 2*(3 + 5*x)]*(11*Sqrt[5] + Sqrt[5]*(3 + 5*x)))/(125*Sqrt[3 + 5*x]) + (66*Sqrt[2/5]*ArcTan[(Sqrt[
2]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/25

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fricas [A]  time = 1.29, size = 82, normalized size = 1.11 \begin {gather*} \frac {33 \, \sqrt {5} \sqrt {2} {\left (5 \, x + 3\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 ) - 20 \, {\left (5 \, x + 14\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{250 \, {\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)^(3/2),x, algorithm="fricas")

[Out]

1/250*(33*sqrt(5)*sqrt(2)*(5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^
2 + x - 3)) - 20*(5*x + 14)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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giac [A]  time = 1.06, size = 98, normalized size = 1.32 \begin {gather*} -\frac {2}{125} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {33}{125} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{250 \, \sqrt {5 \, x + 3}} + \frac {22 \, \sqrt {10} \sqrt {5 \, x + 3}}{125 \, {\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)^(3/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-2/125*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 33/125*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 11/250*sq
rt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 22/125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))

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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 {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.22, size = 62, normalized size = 0.84 \begin {gather*} -\frac {33}{250} \, \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}}}{5 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {33 \, \sqrt {-10 \, x^{2} - x + 3}}{25 \, {\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)^(3/2),x, algorithm="maxima")

[Out]

-33/250*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/5*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) - 33/25*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 )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.56, size = 187, normalized size = 2.53 \begin {gather*} \begin {cases} - \frac {4 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{5 \sqrt {10 x - 5}} - \frac {22 i \sqrt {x + \frac {3}{5}}}{25 \sqrt {10 x - 5}} + \frac {33 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{125} + \frac {242 i}{125 \sqrt {x + \frac {3}{5}} \sqrt {10 x - 5}} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\- \frac {33 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{125} + \frac {4 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{5 \sqrt {5 - 10 x}} + \frac {22 \sqrt {x + \frac {3}{5}}}{25 \sqrt {5 - 10 x}} - \frac {242}{125 \sqrt {5 - 10 x} \sqrt {x + \frac {3}{5}}} & \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)**(3/2),x)

[Out]

Piecewise((-4*I*(x + 3/5)**(3/2)/(5*sqrt(10*x - 5)) - 22*I*sqrt(x + 3/5)/(25*sqrt(10*x - 5)) + 33*sqrt(10)*I*a
cosh(sqrt(110)*sqrt(x + 3/5)/11)/125 + 242*I/(125*sqrt(x + 3/5)*sqrt(10*x - 5)), 10*Abs(x + 3/5)/11 > 1), (-33
*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/125 + 4*(x + 3/5)**(3/2)/(5*sqrt(5 - 10*x)) + 22*sqrt(x + 3/5)/(25*
sqrt(5 - 10*x)) - 242/(125*sqrt(5 - 10*x)*sqrt(x + 3/5)), True))

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