3.27.2 \(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\) [2602]

Optimal. Leaf size=96 \[ -\frac {125}{8} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {25 (3+5 x)^{3/2}}{6 \sqrt {1-2 x}}+\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {275}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]

[Out]

1/3*(3+5*x)^(5/2)/(1-2*x)^(3/2)+275/16*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-25/6*(3+5*x)^(3/2)/(1-2*x)
^(1/2)-125/8*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {49, 52, 56, 222} \begin {gather*} \frac {275}{8} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {(5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {25 (5 x+3)^{3/2}}{6 \sqrt {1-2 x}}-\frac {125}{8} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-125*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8 - (25*(3 + 5*x)^(3/2))/(6*Sqrt[1 - 2*x]) + (3 + 5*x)^(5/2)/(3*(1 - 2*x)^(
3/2)) + (275*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/8

Rule 49

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 52

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 56

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 222

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)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {25}{6} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {25 (3+5 x)^{3/2}}{6 \sqrt {1-2 x}}+\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {125}{4} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {125}{8} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {25 (3+5 x)^{3/2}}{6 \sqrt {1-2 x}}+\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1375}{16} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {125}{8} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {25 (3+5 x)^{3/2}}{6 \sqrt {1-2 x}}+\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1}{8} \left (275 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {125}{8} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {25 (3+5 x)^{3/2}}{6 \sqrt {1-2 x}}+\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {275}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 71, normalized size = 0.74 \begin {gather*} \frac {1}{24} \left (\frac {\sqrt {3+5 x} \left (-603+1840 x-300 x^2\right )}{(1-2 x)^{3/2}}-825 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[3 + 5*x]*(-603 + 1840*x - 300*x^2))/(1 - 2*x)^(3/2) - 825*Sqrt[10]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - Sq
rt[5 - 10*x])])/24

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (3+5 x \right )^{\frac {5}{2}}}{\left (1-2 x \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, size = 129, normalized size = 1.34 \begin {gather*} \frac {275}{32} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {55 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{24 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {605 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1925 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

275/32*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/2*(-10*x^2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x +
1) - 55/24*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 605/48*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1)
+ 1925/48*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]
time = 0.52, size = 97, normalized size = 1.01 \begin {gather*} -\frac {825 \, \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 (300 \, x^{2} - 1840 \, x + 603\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{96 \, {\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)^(5/2)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/96*(825*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*(300*x^2 - 1840*x + 603)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 5.14, size = 728, normalized size = 7.58 \begin {gather*} \begin {cases} - \frac {16500 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} - 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} + \frac {8250 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5}}{960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} - 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} + \frac {18150 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} - 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} - \frac {9075 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}}{960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} - 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} - \frac {30000 i \left (x + \frac {3}{5}\right )^{15}}{960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} - 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} + \frac {220000 i \left (x + \frac {3}{5}\right )^{14}}{960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} - 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} - \frac {181500 i \left (x + \frac {3}{5}\right )^{13}}{960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} - 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {8250 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} - \frac {9075 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} + \frac {15000 \left (x + \frac {3}{5}\right )^{15}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} - \frac {110000 \left (x + \frac {3}{5}\right )^{14}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} + \frac {90750 \left (x + \frac {3}{5}\right )^{13}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-16500*sqrt(10)*I*(x + 3/5)**(27/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(960*(x + 3/5)
**(27/2)*sqrt(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) + 8250*sqrt(10)*pi*(x + 3/5)**(27/2)*sqrt(10*
x - 5)/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) + 18150*sqrt(10)*I*(x +
3/5)**(25/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(x
+ 3/5)**(25/2)*sqrt(10*x - 5)) - 9075*sqrt(10)*pi*(x + 3/5)**(25/2)*sqrt(10*x - 5)/(960*(x + 3/5)**(27/2)*sqrt
(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) - 30000*I*(x + 3/5)**15/(960*(x + 3/5)**(27/2)*sqrt(10*x -
5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) + 220000*I*(x + 3/5)**14/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) -
1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) - 181500*I*(x + 3/5)**13/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(
x + 3/5)**(25/2)*sqrt(10*x - 5)), Abs(x + 3/5) > 11/10), (8250*sqrt(10)*sqrt(5 - 10*x)*(x + 3/5)**(27/2)*asin(
sqrt(110)*sqrt(x + 3/5)/11)/(480*sqrt(5 - 10*x)*(x + 3/5)**(27/2) - 528*sqrt(5 - 10*x)*(x + 3/5)**(25/2)) - 90
75*sqrt(10)*sqrt(5 - 10*x)*(x + 3/5)**(25/2)*asin(sqrt(110)*sqrt(x + 3/5)/11)/(480*sqrt(5 - 10*x)*(x + 3/5)**(
27/2) - 528*sqrt(5 - 10*x)*(x + 3/5)**(25/2)) + 15000*(x + 3/5)**15/(480*sqrt(5 - 10*x)*(x + 3/5)**(27/2) - 52
8*sqrt(5 - 10*x)*(x + 3/5)**(25/2)) - 110000*(x + 3/5)**14/(480*sqrt(5 - 10*x)*(x + 3/5)**(27/2) - 528*sqrt(5
- 10*x)*(x + 3/5)**(25/2)) + 90750*(x + 3/5)**13/(480*sqrt(5 - 10*x)*(x + 3/5)**(27/2) - 528*sqrt(5 - 10*x)*(x
+ 3/5)**(25/2)), True))

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Giac [A]
time = 1.37, size = 71, normalized size = 0.74 \begin {gather*} \frac {275}{16} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, \sqrt {5} {\left (5 \, x + 3\right )} - 110 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1815 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{120 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

275/16*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/120*(4*(3*sqrt(5)*(5*x + 3) - 110*sqrt(5))*(5*x + 3) +
1815*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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

[Out]

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

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