3.25.68 \(\int \frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\) [2468]

Optimal. Leaf size=128 \[ -\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{512000 \sqrt {10}} \]

[Out]

333216939/5120000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-917953/128000*(3+5*x)^(3/2)*(1-2*x)^(1/2)-3/50*
(2+3*x)^2*(3+5*x)^(5/2)*(1-2*x)^(1/2)-3/16000*(3+5*x)^(5/2)*(7889+3900*x)*(1-2*x)^(1/2)-30292449/512000*(1-2*x
)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {333216939 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{512000 \sqrt {10}}-\frac {3}{50} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}-\frac {3 \sqrt {1-2 x} (3900 x+7889) (5 x+3)^{5/2}}{16000}-\frac {917953 \sqrt {1-2 x} (5 x+3)^{3/2}}{128000}-\frac {30292449 \sqrt {1-2 x} \sqrt {5 x+3}}{512000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-30292449*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/512000 - (917953*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/128000 - (3*Sqrt[1 - 2
*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/50 - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(7889 + 3900*x))/16000 + (333216939*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(512000*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 {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {1}{50} \int \frac {\left (-311-\frac {975 x}{2}\right ) (2+3 x) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {917953 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{32000}\\ &=-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {30292449 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{256000}\\ &=-\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1024000}\\ &=-\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{512000 \sqrt {5}}\\ &=-\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{512000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 83, normalized size = 0.65 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (147689703+400508925 x+397621060 x^2+314536800 x^3+155088000 x^4+34560000 x^5\right )-333216939 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{5120000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*(147689703 + 400508925*x + 397621060*x^2 + 314536800*x^3 + 155088000*x^4 + 34560000*x^5) -
333216939*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(5120000*Sqrt[3 + 5*x])

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Maple [A]
time = 0.08, size = 121, normalized size = 0.95

method	result	size
risch	\(\frac {\left (6912000 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{512000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {333216939 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{10240000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(108\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-138240000 x^{4} \sqrt {-10 x^{2}-x +3}-537408000 x^{3} \sqrt {-10 x^{2}-x +3}-935702400 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1029062800 x \sqrt {-10 x^{2}-x +3}-984598020 \sqrt {-10 x^{2}-x +3}\right )}{10240000 \sqrt {-10 x^{2}-x +3}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10240000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-138240000*x^4*(-10*x^2-x+3)^(1/2)-537408000*x^3*(-10*x^2-x+3)^(1/2)-9
35702400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-1029062800*x*(-10*x^2-x+3)^(1/2)-9845
98020*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.52, size = 92, normalized size = 0.72 \begin {gather*} -\frac {27}{2} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {8397}{160} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {292407}{3200} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {2572657}{25600} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {333216939}{10240000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {49229901}{512000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/2*sqrt(-10*x^2 - x + 3)*x^4 - 8397/160*sqrt(-10*x^2 - x + 3)*x^3 - 292407/3200*sqrt(-10*x^2 - x + 3)*x^2 -
2572657/25600*sqrt(-10*x^2 - x + 3)*x - 333216939/10240000*sqrt(10)*arcsin(-20/11*x - 1/11) - 49229901/512000
*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 0.39, size = 77, normalized size = 0.60 \begin {gather*} -\frac {1}{512000} \, {\left (6912000 \, x^{4} + 26870400 \, x^{3} + 46785120 \, x^{2} + 51453140 \, x + 49229901\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {333216939}{10240000} \, \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((2+3*x)^3*(3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/512000*(6912000*x^4 + 26870400*x^3 + 46785120*x^2 + 51453140*x + 49229901)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3
33216939/10240000*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 [A]
time = 162.82, size = 678, normalized size = 5.30 \begin {gather*} \frac {2 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (\frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{968} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{8}\right )}{8} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{625} + \frac {18 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (\frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {3 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{16} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{625} + \frac {54 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \cdot \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} + \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{32} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{625} + \frac {54 \sqrt {5} \left (\begin {cases} \frac {161051 \sqrt {2} \left (- \frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} + \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{1331} + \frac {15 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{7744} + \frac {5 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{3748096} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {63 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{256}\right )}{64} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/968 - sqrt(2)*sqrt(5 - 10*x
)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) <
sqrt(22)/2)))/625 + 18*sqrt(5)*Piecewise((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3*sq
rt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 5*asin(sqrt(22
)*sqrt(5*x + 3)/11)/16)/16, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/625 + 54*sqrt(5)*Pi
ecewise((14641*sqrt(2)*(2*sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 7*sqrt(2)*sqrt(5 - 10*x)*(-20*x -
1)*sqrt(5*x + 3)/3872 + sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2
- 5929)/1874048 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 35*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)/32, (sqrt(
5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/625 + 54*sqrt(5)*Piecewise((161051*sqrt(2)*(-2*sqrt(2
)*(5 - 10*x)**(5/2)*(5*x + 3)**(5/2)/805255 + sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/1331 + 15*sqrt(2)*sqr
t(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/7744 + 5*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)*
*3 + 1056*(5*x + 3)**2 - 5929)/3748096 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 63*asin(sqrt(22)*sqrt(5*x +
3)/11)/256)/64, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/625

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Giac [A]
time = 1.60, size = 72, normalized size = 0.56 \begin {gather*} -\frac {1}{25600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (24 \, {\left (36 \, {\left (80 \, x + 167\right )} {\left (5 \, x + 3\right )} + 27809\right )} {\left (5 \, x + 3\right )} + 4589765\right )} {\left (5 \, x + 3\right )} + 151462245\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 1666084695 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/25600000*sqrt(5)*(2*(4*(24*(36*(80*x + 167)*(5*x + 3) + 27809)*(5*x + 3) + 4589765)*(5*x + 3) + 151462245)*
sqrt(5*x + 3)*sqrt(-10*x + 5) - 1666084695*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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