3.25.49 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx\) [2449]

Optimal. Leaf size=138 \[ -\frac {2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac {76 (1-2 x)^{7/2}}{1815 \sqrt {3+5 x}}+\frac {329 \sqrt {1-2 x} \sqrt {3+5 x}}{5000}+\frac {329 (1-2 x)^{3/2} \sqrt {3+5 x}}{16500}+\frac {329 (1-2 x)^{5/2} \sqrt {3+5 x}}{45375}+\frac {3619 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5000 \sqrt {10}} \]

[Out]

-2/825*(1-2*x)^(7/2)/(3+5*x)^(3/2)+3619/50000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-76/1815*(1-2*x)^(7/
2)/(3+5*x)^(1/2)+329/16500*(1-2*x)^(3/2)*(3+5*x)^(1/2)+329/45375*(1-2*x)^(5/2)*(3+5*x)^(1/2)+329/5000*(1-2*x)^
(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {91, 79, 52, 56, 222} \begin {gather*} \frac {3619 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5000 \sqrt {10}}-\frac {76 (1-2 x)^{7/2}}{1815 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}+\frac {329 \sqrt {5 x+3} (1-2 x)^{5/2}}{45375}+\frac {329 \sqrt {5 x+3} (1-2 x)^{3/2}}{16500}+\frac {329 \sqrt {5 x+3} \sqrt {1-2 x}}{5000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(7/2))/(825*(3 + 5*x)^(3/2)) - (76*(1 - 2*x)^(7/2))/(1815*Sqrt[3 + 5*x]) + (329*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/5000 + (329*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/16500 + (329*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/45375 + (3
619*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5000*Sqrt[10])

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 79

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] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}+\frac {2}{825} \int \frac {(1-2 x)^{5/2} \left (\frac {1081}{2}+\frac {1485 x}{2}\right )}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac {76 (1-2 x)^{7/2}}{1815 \sqrt {3+5 x}}+\frac {329 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{3025}\\ &=-\frac {2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac {76 (1-2 x)^{7/2}}{1815 \sqrt {3+5 x}}+\frac {329 (1-2 x)^{5/2} \sqrt {3+5 x}}{45375}+\frac {329 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{1650}\\ &=-\frac {2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac {76 (1-2 x)^{7/2}}{1815 \sqrt {3+5 x}}+\frac {329 (1-2 x)^{3/2} \sqrt {3+5 x}}{16500}+\frac {329 (1-2 x)^{5/2} \sqrt {3+5 x}}{45375}+\frac {329 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1000}\\ &=-\frac {2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac {76 (1-2 x)^{7/2}}{1815 \sqrt {3+5 x}}+\frac {329 \sqrt {1-2 x} \sqrt {3+5 x}}{5000}+\frac {329 (1-2 x)^{3/2} \sqrt {3+5 x}}{16500}+\frac {329 (1-2 x)^{5/2} \sqrt {3+5 x}}{45375}+\frac {3619 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{10000}\\ &=-\frac {2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac {76 (1-2 x)^{7/2}}{1815 \sqrt {3+5 x}}+\frac {329 \sqrt {1-2 x} \sqrt {3+5 x}}{5000}+\frac {329 (1-2 x)^{3/2} \sqrt {3+5 x}}{16500}+\frac {329 (1-2 x)^{5/2} \sqrt {3+5 x}}{45375}+\frac {3619 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac {76 (1-2 x)^{7/2}}{1815 \sqrt {3+5 x}}+\frac {329 \sqrt {1-2 x} \sqrt {3+5 x}}{5000}+\frac {329 (1-2 x)^{3/2} \sqrt {3+5 x}}{16500}+\frac {329 (1-2 x)^{5/2} \sqrt {3+5 x}}{45375}+\frac {3619 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 82, normalized size = 0.59 \begin {gather*} \frac {\frac {5 \sqrt {1-2 x} \left (10633+40930 x+3585 x^2-35100 x^3+36000 x^4\right )}{(3+5 x)^{3/2}}-10857 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{75000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((5*Sqrt[1 - 2*x]*(10633 + 40930*x + 3585*x^2 - 35100*x^3 + 36000*x^4))/(3 + 5*x)^(3/2) - 10857*Sqrt[10]*ArcTa
n[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x])])/75000

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Maple [A]
time = 0.25, size = 147, normalized size = 1.07

method	result	size
default	\(\frac {\left (720000 x^{4} \sqrt {-10 x^{2}-x +3}+271425 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-702000 x^{3} \sqrt {-10 x^{2}-x +3}+325710 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +71700 x^{2} \sqrt {-10 x^{2}-x +3}+97713 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+818600 x \sqrt {-10 x^{2}-x +3}+212660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{300000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(147\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/300000*(720000*x^4*(-10*x^2-x+3)^(1/2)+271425*10^(1/2)*arcsin(20/11*x+1/11)*x^2-702000*x^3*(-10*x^2-x+3)^(1/
2)+325710*10^(1/2)*arcsin(20/11*x+1/11)*x+71700*x^2*(-10*x^2-x+3)^(1/2)+97713*10^(1/2)*arcsin(20/11*x+1/11)+81
8600*x*(-10*x^2-x+3)^(1/2)+212660*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (99) = 198\).
time = 0.57, size = 247, normalized size = 1.79 \begin {gather*} \frac {3619}{100000} \, \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}}}{125 \, {\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}}}{125 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {1089}{5000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {11 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{750 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{500 \, {\left (5 \, x + 3\right )}} - \frac {121 \, \sqrt {-10 \, x^{2} - x + 3}}{3750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {3113 \, \sqrt {-10 \, x^{2} - x + 3}}{3750 \, {\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)^2/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

3619/100000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/125*(-10*x^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*
x^2 + 540*x + 81) + 3/125*(-10*x^2 - x + 3)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 3/125*(-10*x^2 - x + 3)^(
5/2)/(25*x^2 + 30*x + 9) + 1089/5000*sqrt(-10*x^2 - x + 3) - 11/750*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2
+ 135*x + 27) + 33/250*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 33/500*(-10*x^2 - x + 3)^(3/2)/(5*x + 3)
- 121/3750*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 3113/3750*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]
time = 0.43, size = 101, normalized size = 0.73 \begin {gather*} -\frac {10857 \, \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 (36000 \, x^{4} - 35100 \, x^{3} + 3585 \, x^{2} + 40930 \, x + 10633\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{300000 \, {\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)^2/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/300000*(10857*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*(36000*x^4 - 35100*x^3 + 3585*x^2 + 40930*x + 10633)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2
+ 30*x + 9)

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Sympy [F(-1)] Timed out
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)**2/(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [A]
time = 0.68, size = 184, normalized size = 1.33 \begin {gather*} \frac {1}{125000} \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 135 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9635 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11}{750000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {1476 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {3619}{50000} \, \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 {369 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{46875 \, {\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)^2/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

1/125000*(12*(8*sqrt(5)*(5*x + 3) - 135*sqrt(5))*(5*x + 3) + 9635*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/
750000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 1476*(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))/sqrt(5*x + 3)) + 3619/50000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 11/46875*sqrt(10)*(5*x + 3)^(
3/2)*(369*(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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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 )}^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)^(5/2)*(3*x + 2)^2)/(5*x + 3)^(5/2),x)

[Out]

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

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