3.2337 \(\int (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=172 \[ -\frac{3}{70} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{141599 (5 x+3)^{3/2} (1-2 x)^{5/2}}{128000}-\frac{3 (5 x+3)^{5/2} (33300 x+49829) (1-2 x)^{5/2}}{280000}-\frac{1557589 \sqrt{5 x+3} (1-2 x)^{5/2}}{512000}+\frac{17133479 \sqrt{5 x+3} (1-2 x)^{3/2}}{10240000}+\frac{565404807 \sqrt{5 x+3} \sqrt{1-2 x}}{102400000}+\frac{6219452877 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400000 \sqrt{10}} \]

[Out]

(565404807*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400000 + (17133479*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10240000 - (15575
89*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/512000 - (141599*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/128000 - (3*(1 - 2*x)^(5/2
)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/70 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)*(49829 + 33300*x))/280000 + (6219452877
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400000*Sqrt[10])

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Rubi [A]  time = 0.0510895, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac{3}{70} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{141599 (5 x+3)^{3/2} (1-2 x)^{5/2}}{128000}-\frac{3 (5 x+3)^{5/2} (33300 x+49829) (1-2 x)^{5/2}}{280000}-\frac{1557589 \sqrt{5 x+3} (1-2 x)^{5/2}}{512000}+\frac{17133479 \sqrt{5 x+3} (1-2 x)^{3/2}}{10240000}+\frac{565404807 \sqrt{5 x+3} \sqrt{1-2 x}}{102400000}+\frac{6219452877 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(565404807*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400000 + (17133479*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10240000 - (15575
89*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/512000 - (141599*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/128000 - (3*(1 - 2*x)^(5/2
)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/70 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)*(49829 + 33300*x))/280000 + (6219452877
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400000*Sqrt[10])

Rule 100

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 147

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 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 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 (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx &=-\frac{3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{1}{70} \int \left (-319-\frac{999 x}{2}\right ) (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx\\ &=-\frac{3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac{141599 \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx}{16000}\\ &=-\frac{141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac{4672767 \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx}{256000}\\ &=-\frac{1557589 (1-2 x)^{5/2} \sqrt{3+5 x}}{512000}-\frac{141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac{17133479 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{1024000}\\ &=\frac{17133479 (1-2 x)^{3/2} \sqrt{3+5 x}}{10240000}-\frac{1557589 (1-2 x)^{5/2} \sqrt{3+5 x}}{512000}-\frac{141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac{565404807 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{20480000}\\ &=\frac{565404807 \sqrt{1-2 x} \sqrt{3+5 x}}{102400000}+\frac{17133479 (1-2 x)^{3/2} \sqrt{3+5 x}}{10240000}-\frac{1557589 (1-2 x)^{5/2} \sqrt{3+5 x}}{512000}-\frac{141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac{6219452877 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{204800000}\\ &=\frac{565404807 \sqrt{1-2 x} \sqrt{3+5 x}}{102400000}+\frac{17133479 (1-2 x)^{3/2} \sqrt{3+5 x}}{10240000}-\frac{1557589 (1-2 x)^{5/2} \sqrt{3+5 x}}{512000}-\frac{141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac{6219452877 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{102400000 \sqrt{5}}\\ &=\frac{565404807 \sqrt{1-2 x} \sqrt{3+5 x}}{102400000}+\frac{17133479 (1-2 x)^{3/2} \sqrt{3+5 x}}{10240000}-\frac{1557589 (1-2 x)^{5/2} \sqrt{3+5 x}}{512000}-\frac{141599 (1-2 x)^{5/2} (3+5 x)^{3/2}}{128000}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 (1-2 x)^{5/2} (3+5 x)^{5/2} (49829+33300 x)}{280000}+\frac{6219452877 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{102400000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.190544, size = 89, normalized size = 0.52 \[ \frac{10 \sqrt{5 x+3} \left (55296000000 x^7+108288000000 x^6+25496064000 x^5-71786259200 x^4-44888000960 x^3+10130684360 x^2+17193258662 x-3952411101\right )-43536170139 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{7168000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-3952411101 + 17193258662*x + 10130684360*x^2 - 44888000960*x^3 - 71786259200*x^4 + 2549606
4000*x^5 + 108288000000*x^6 + 55296000000*x^7) - 43536170139*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])
/(7168000000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.01, size = 155, normalized size = 0.9 \begin{align*}{\frac{1}{14336000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -552960000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}-1359360000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-934640640000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-79048222020\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14336000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-552960000000*(-10*x^2-x+3)^(1/2)*x^6-1359360000000*x^5*(-10*x^2-x+
3)^(1/2)-934640640000*x^4*(-10*x^2-x+3)^(1/2)[phone]*x^3*(-10*x^2-x+3)^(1/2)[phone]*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)-79048222020*(-10*x^2-x+
3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.14203, size = 157, normalized size = 0.91 \begin{align*} -\frac{27}{70} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} - \frac{2439}{2800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{197487}{280000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{141599}{64000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{141599}{1280000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{51400437}{5120000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{6219452877}{2048000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{51400437}{102400000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/70*(-10*x^2 - x + 3)^(5/2)*x^2 - 2439/2800*(-10*x^2 - x + 3)^(5/2)*x - 197487/280000*(-10*x^2 - x + 3)^(5/
2) + 141599/64000*(-10*x^2 - x + 3)^(3/2)*x + 141599/1280000*(-10*x^2 - x + 3)^(3/2) + 51400437/5120000*sqrt(-
10*x^2 - x + 3)*x - 6219452877/2048000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 51400437/102400000*sqrt(-10*x^2 -
x + 3)

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Fricas [A]  time = 1.49385, size = 369, normalized size = 2.15 \begin{align*} -\frac{1}{716800000} \,{\left (27648000000 \, x^{6} + 67968000000 \, x^{5} + 46732032000 \, x^{4} - 12527113600 \, x^{3} - 28707557280 \, x^{2} - 9288436460 \, x + 3952411101\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{6219452877}{2048000000} \, \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/716800000*(27648000000*x^6 + 67968000000*x^5 + 46732032000*x^4 - 12527113600*x^3 - 28707557280*x^2 - 928843
6460*x + 3952411101)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 6219452877/2048000000*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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.27035, size = 548, normalized size = 3.19 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/35840000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x + 3) - 3968469)*(5*x + 3) + 3
3617829)*(5*x + 3) - 276044685)*(5*x + 3) + 87356115)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 960917265*sqrt(2)*arcsin
(1/11*sqrt(22)*sqrt(5*x + 3))) - 189/2560000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999)*(5*x
+ 3) - 318159)*(5*x + 3) + 3237255)*(5*x + 3) - 2656665)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 29223315*sqrt(2)*arcs
in(1/11*sqrt(22)*sqrt(5*x + 3))) - 111/64000000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3)
- 136405)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)
)) + 23/960000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5)
+ 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/240*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*
x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/50*sqrt(5)*(2*(20*x + 1)*sqrt(5*
x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))