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

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

________________________________________________________________________________________

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 {(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {5 \sqrt {5 x+3}}{2 \sqrt {1-2 x}}+\frac {5}{2} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]) + (3 + 5*x)^(3/2)/(3*(1 - 2*x)^(3/2)) + (5*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]])/2

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 {(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {5}{2} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {5 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {25}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {5 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {1}{2} \left (5 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {5 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {5}{2} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 39, normalized size = 0.53 \begin {gather*} \frac {11 \sqrt {\frac {11}{2}} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )}{6 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11*Sqrt[11/2]*Hypergeometric2F1[-3/2, -3/2, -1/2, (-5*(-1 + 2*x))/11])/(6*(1 - 2*x)^(3/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.09, size = 77, normalized size = 1.04 \begin {gather*} \frac {(5 x+3)^{3/2} \left (2-\frac {15 (1-2 x)}{5 x+3}\right )}{6 (1-2 x)^{3/2}}-\frac {5}{2} \sqrt {\frac {5}{2}} \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[(3 + 5*x)^(3/2)/(1 - 2*x)^(5/2),x]

[Out]

((3 + 5*x)^(3/2)*(2 - (15*(1 - 2*x))/(3 + 5*x)))/(6*(1 - 2*x)^(3/2)) - (5*Sqrt[5/2]*ArcTan[(Sqrt[5/2]*Sqrt[1 -
2*x])/Sqrt[3 + 5*x]])/2

________________________________________________________________________________________

fricas [A]  time = 0.86, size = 92, normalized size = 1.24 \begin {gather*} -\frac {15 \, \sqrt {5} \sqrt {2} {\left (4 \, x^{2} - 4 \, x + 1\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 ) - 4 \, {\left (40 \, x - 9\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{24 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(15*sqrt(5)*sqrt(2)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1
)/(10*x^2 + x - 3)) - 4*(40*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

________________________________________________________________________________________

giac [A]  time = 1.30, size = 58, normalized size = 0.78 \begin {gather*} \frac {5}{4} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{30 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/30*(8*sqrt(5)*(5*x + 3) - 33*sqrt(5))*sqrt(5*x + 3)*sqrt(
-10*x + 5)/(2*x - 1)^2

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.22, size = 93, normalized size = 1.26 \begin {gather*} \frac {5}{8} \, \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}}}{6 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {11 \, \sqrt {-10 \, x^{2} - x + 3}}{12 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {35 \, \sqrt {-10 \, x^{2} - x + 3}}{12 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/8*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/6*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 11/12*sq
rt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 35/12*sqrt(-10*x^2 - x + 3)/(2*x - 1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 3.87, size = 636, normalized size = 8.59 \begin {gather*} \begin {cases} \frac {300 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{- 240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} + 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} - \frac {150 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5}}{- 240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} + 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} - \frac {330 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{- 240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} + 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} + \frac {165 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}}{- 240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} + 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} - \frac {4000 i \left (x + \frac {3}{5}\right )^{8}}{- 240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} + 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} + \frac {3300 i \left (x + \frac {3}{5}\right )^{7}}{- 240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} + 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {150 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} - \frac {165 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} - \frac {2000 \left (x + \frac {3}{5}\right )^{8}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} + \frac {1650 \left (x + \frac {3}{5}\right )^{7}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((300*sqrt(10)*I*(x + 3/5)**(15/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(-240*(x + 3/5)**
(15/2)*sqrt(10*x - 5) + 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) - 150*sqrt(10)*pi*(x + 3/5)**(15/2)*sqrt(10*x -
5)/(-240*(x + 3/5)**(15/2)*sqrt(10*x - 5) + 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) - 330*sqrt(10)*I*(x + 3/5)**
(13/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(-240*(x + 3/5)**(15/2)*sqrt(10*x - 5) + 264*(x + 3/5)
**(13/2)*sqrt(10*x - 5)) + 165*sqrt(10)*pi*(x + 3/5)**(13/2)*sqrt(10*x - 5)/(-240*(x + 3/5)**(15/2)*sqrt(10*x
- 5) + 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) - 4000*I*(x + 3/5)**8/(-240*(x + 3/5)**(15/2)*sqrt(10*x - 5) + 26
4*(x + 3/5)**(13/2)*sqrt(10*x - 5)) + 3300*I*(x + 3/5)**7/(-240*(x + 3/5)**(15/2)*sqrt(10*x - 5) + 264*(x + 3/
5)**(13/2)*sqrt(10*x - 5)), 10*Abs(x + 3/5)/11 > 1), (150*sqrt(10)*sqrt(5 - 10*x)*(x + 3/5)**(15/2)*asin(sqrt(
110)*sqrt(x + 3/5)/11)/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2) - 132*sqrt(5 - 10*x)*(x + 3/5)**(13/2)) - 165*sqr
t(10)*sqrt(5 - 10*x)*(x + 3/5)**(13/2)*asin(sqrt(110)*sqrt(x + 3/5)/11)/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2)
- 132*sqrt(5 - 10*x)*(x + 3/5)**(13/2)) - 2000*(x + 3/5)**8/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2) - 132*sqrt(5
- 10*x)*(x + 3/5)**(13/2)) + 1650*(x + 3/5)**7/(120*sqrt(5 - 10*x)*(x + 3/5)**(15/2) - 132*sqrt(5 - 10*x)*(x
+ 3/5)**(13/2)), True))

________________________________________________________________________________________