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

Optimal. Leaf size=116 \[ -\frac {2 (1-2 x)^{7/2}}{165 (5 x+3)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {5 x+3}}-\frac {91}{825} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {91}{250} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {1001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{250 \sqrt {10}} \]

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Rubi [A]  time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {78, 47, 50, 54, 216} \begin {gather*} -\frac {2 (1-2 x)^{7/2}}{165 (5 x+3)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {5 x+3}}-\frac {91}{825} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {91}{250} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {1001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{250 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(7/2))/(165*(3 + 5*x)^(3/2)) - (182*(1 - 2*x)^(5/2))/(825*Sqrt[3 + 5*x]) - (91*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/250 - (91*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/825 - (1001*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(250*Sqrt[1
0])

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 78

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

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)}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}+\frac {91}{165} \int \frac {(1-2 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {182}{165} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {91}{825} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {91}{50} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {91}{250} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {91}{825} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1001}{500} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {91}{250} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {91}{825} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1001 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{250 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {91}{250} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {91}{825} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{250 \sqrt {10}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 59, normalized size = 0.51 \begin {gather*} -\frac {2 (1-2 x)^{7/2} \left (13 \sqrt {22} (5 x+3)^{3/2} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};-\frac {5}{11} (2 x-1)\right )+121\right )}{19965 (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(7/2)*(121 + 13*Sqrt[22]*(3 + 5*x)^(3/2)*Hypergeometric2F1[3/2, 7/2, 9/2, (-5*(-1 + 2*x))/11]))/
(19965*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.43, size = 123, normalized size = 1.06 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (36 \sqrt {5} (5 x+3)^3-867 \sqrt {5} (5 x+3)^2-3740 \sqrt {5} (5 x+3)-484 \sqrt {5}\right )}{18750 (5 x+3)^{3/2}}+\frac {1001 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{125 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-484*Sqrt[5] - 3740*Sqrt[5]*(3 + 5*x) - 867*Sqrt[5]*(3 + 5*x)^2 + 36*Sqrt[5]*(3 + 5*x
)^3))/(18750*(3 + 5*x)^(3/2)) + (1001*ArcTan[(Sqrt[2]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(12
5*Sqrt[10])

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fricas [A]  time = 1.38, size = 96, normalized size = 0.83 \begin {gather*} \frac {3003 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\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 (900 \, x^{3} - 2715 \, x^{2} - 7970 \, x - 3707\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{15000 \, {\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)^(5/2)*(2+3*x)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

1/15000*(3003*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^
2 + x - 3)) + 20*(900*x^3 - 2715*x^2 - 7970*x - 3707)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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giac [B]  time = 1.86, size = 171, normalized size = 1.47 \begin {gather*} \frac {1}{6250} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 289 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11}{150000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {684 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {1001}{2500} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {11 \, \sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {171 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{9375 \, {\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)^(5/2)*(2+3*x)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

1/6250*(12*sqrt(5)*(5*x + 3) - 289*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/150000*sqrt(10)*((sqrt(2)*sqrt(
-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 684*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3)) - 1001/2500
*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 11/9375*sqrt(10)*(5*x + 3)^(3/2)*(171*(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 [A]  time = 0.01, size = 130, normalized size = 1.12 \begin {gather*} -\frac {\left (-18000 \sqrt {-10 x^{2}-x +3}\, x^{3}+75075 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+54300 \sqrt {-10 x^{2}-x +3}\, x^{2}+90090 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+159400 \sqrt {-10 x^{2}-x +3}\, x +27027 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+74140 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{15000 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15000*(75075*10^(1/2)*x^2*arcsin(20/11*x+1/11)-18000*(-10*x^2-x+3)^(1/2)*x^3+90090*10^(1/2)*x*arcsin(20/11*
x+1/11)+54300*(-10*x^2-x+3)^(1/2)*x^2+27027*10^(1/2)*arcsin(20/11*x+1/11)+159400*(-10*x^2-x+3)^(1/2)*x+74140*(
-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(3/2)

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maxima [B]  time = 1.24, size = 186, normalized size = 1.60 \begin {gather*} -\frac {1001}{5000} \, \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}}}{25 \, {\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{50 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} - \frac {11 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{150 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{100 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {121 \, \sqrt {-10 \, x^{2} - x + 3}}{750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {2959 \, \sqrt {-10 \, x^{2} - x + 3}}{1500 \, {\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)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

-1001/5000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/25*(-10*x^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*x^
2 + 540*x + 81) + 3/50*(-10*x^2 - x + 3)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) - 11/150*(-10*x^2 - x + 3)^(3/
2)/(125*x^3 + 225*x^2 + 135*x + 27) + 33/100*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) - 121/750*sqrt(-10*x^
2 - x + 3)/(25*x^2 + 30*x + 9) - 2959/1500*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 )}^{5/2}\,\left (3\,x+2\right )}{{\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)^(5/2)*(3*x + 2))/(5*x + 3)^(5/2),x)

[Out]

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

[Out]

Timed out

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