3.25.25 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx\) [2425]

Optimal. Leaf size=150 \[ \frac {33455653 \sqrt {1-2 x} \sqrt {3+5 x}}{64000000}+\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {368012183 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000000 \sqrt {10}} \]

[Out]

368012183/640000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+3041423/19200000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+
276493/4800000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-1/20*(1-2*x)^(7/2)*(2+3*x)^2*(3+5*x)^(1/2)-1/160000*(1-2*x)^(7/2)*(
52951+47280*x)*(3+5*x)^(1/2)[phone]/64000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 150, 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 = {102, 152, 52, 56, 222} \begin {gather*} \frac {368012183 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64000000 \sqrt {10}}-\frac {\sqrt {5 x+3} (47280 x+52951) (1-2 x)^{7/2}}{160000}-\frac {1}{20} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {276493 \sqrt {5 x+3} (1-2 x)^{5/2}}{4800000}+\frac {3041423 \sqrt {5 x+3} (1-2 x)^{3/2}}{19200000}+\frac {33455653 \sqrt {5 x+3} \sqrt {1-2 x}}{64000000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(33455653*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000000 + (3041423*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/19200000 + (276493*(
1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/4800000 - ((1 - 2*x)^(7/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/20 - ((1 - 2*x)^(7/2)*Sqrt
[3 + 5*x]*(52951 + 47280*x))/160000 + (368012183*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000000*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 102

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

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx &=-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{60} \int \frac {\left (-183-\frac {591 x}{2}\right ) (1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx\\ &=-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {276493 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{320000}\\ &=\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {3041423 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{1920000}\\ &=\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {33455653 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{12800000}\\ &=\frac {33455653 \sqrt {1-2 x} \sqrt {3+5 x}}{64000000}+\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {368012183 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{128000000}\\ &=\frac {33455653 \sqrt {1-2 x} \sqrt {3+5 x}}{64000000}+\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {368012183 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{64000000 \sqrt {5}}\\ &=\frac {33455653 \sqrt {1-2 x} \sqrt {3+5 x}}{64000000}+\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {368012183 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 88, normalized size = 0.59 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (119699127+1203430125 x+971700740 x^2-3357104800 x^3-2630448000 x^4+3767040000 x^5+3456000000 x^6\right )-1104036549 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1920000000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(119699127 + 1203430125*x + 971700740*x^2 - 3357104800*x^3 - 2630448000*x^4 + 3767040000*x^5
+ 3456000000*x^6) - 1104036549*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(1920000000*Sqrt[3 + 5*
x])

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Maple [A]
time = 0.10, size = 138, normalized size = 0.92

method	result	size
risch	\(-\frac {\left (691200000 x^{5}[phone] x^{4}-729302400 x^{3}-233839520 x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{192000000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {368012183 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1280000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(113\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (13824000000 x^{5} \sqrt {-10 x^{2}-x +3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}-14586048000 x^{3} \sqrt {-10 x^{2}-x +3}-4676790400 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{3840000000 \sqrt {-10 x^{2}-x +3}}\)	\(138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13824000000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/
2)-14586048000*x^3*(-10*x^2-x+3)^(1/2)-4676790400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1
/11)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.55, size = 109, normalized size = 0.73 \begin {gather*} \frac {18}{5} \, \sqrt {-10 \, x^{2} - x + 3} x^{5} + \frac {441}{250} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {75969}{20000} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {1461497}{1200000} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {16732193}{9600000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {368012183}{1280000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {13299903}{64000000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18/5*sqrt(-10*x^2 - x + 3)*x^5 + 441/250*sqrt(-10*x^2 - x + 3)*x^4 - 75969/20000*sqrt(-10*x^2 - x + 3)*x^3 - 1
461497/1200000*sqrt(-10*x^2 - x + 3)*x^2 + 16732193/9600000*sqrt(-10*x^2 - x + 3)*x - 368012183/1280000000*sqr
t(10)*arcsin(-20/11*x - 1/11) + 13299903/64000000*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 0.49, size = 82, normalized size = 0.55 \begin {gather*} \frac {1}{192000000} \, {\left (691200000 \, x^{5} + 338688000 \, x^{4} - 729302400 \, x^{3} - 233839520 \, x^{2} + 334643860 \, x + 39899709\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {368012183}{1280000000} \, \sqrt {10} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192000000*(691200000*x^5 + 338688000*x^4 - 729302400*x^3 - 233839520*x^2 + 334643860*x + 39899709)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) - 368012183/1280000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3))

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (111) = 222\).
time = 0.61, size = 356, normalized size = 2.37 \begin {gather*} \frac {9}{3200000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{80000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {3}{640000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {29}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {4}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/3200000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 15385695
)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
+ 9/80000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sq
rt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 3/640000*sqrt(5)*(2*(4*(
8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3))) - 29/60000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5)
+ 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/500*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x +
5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 4/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*
x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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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 )}^3}{\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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