3.2350 \(\int (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=187 \[ -\frac{47}{400} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{7/2}-\frac{783 (1-2 x)^{5/2} (5 x+3)^{5/2}}{1600}-\frac{8613 (1-2 x)^{5/2} (5 x+3)^{3/2}}{5120}-\frac{94743 (1-2 x)^{5/2} \sqrt{5 x+3}}{20480}+\frac{1042173 (1-2 x)^{3/2} \sqrt{5 x+3}}{409600}+\frac{34391709 \sqrt{1-2 x} \sqrt{5 x+3}}{4096000}+\frac{378308799 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4096000 \sqrt{10}} \]

[Out]

(34391709*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096000 + (1042173*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/409600 - (94743*(1 -
2*x)^(5/2)*Sqrt[3 + 5*x])/20480 - (8613*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/5120 - (783*(1 - 2*x)^(5/2)*(3 + 5*x)
^(5/2))/1600 - (47*(1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/400 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/70 + (
378308799*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096000*Sqrt[10])

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Rubi [A]  time = 0.0614622, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac{47}{400} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{7/2}-\frac{783 (1-2 x)^{5/2} (5 x+3)^{5/2}}{1600}-\frac{8613 (1-2 x)^{5/2} (5 x+3)^{3/2}}{5120}-\frac{94743 (1-2 x)^{5/2} \sqrt{5 x+3}}{20480}+\frac{1042173 (1-2 x)^{3/2} \sqrt{5 x+3}}{409600}+\frac{34391709 \sqrt{1-2 x} \sqrt{5 x+3}}{4096000}+\frac{378308799 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4096000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(34391709*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096000 + (1042173*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/409600 - (94743*(1 -
2*x)^(5/2)*Sqrt[3 + 5*x])/20480 - (8613*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/5120 - (783*(1 - 2*x)^(5/2)*(3 + 5*x)
^(5/2))/1600 - (47*(1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/400 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/70 + (
378308799*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096000*Sqrt[10])

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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)^2 (3+5 x)^{5/2} \, dx &=-\frac{3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}-\frac{1}{70} \int \left (-322-\frac{987 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx\\ &=-\frac{47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac{783}{160} \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx\\ &=-\frac{783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac{47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac{8613}{640} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac{783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac{47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac{284229 \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx}{10240}\\ &=-\frac{94743 (1-2 x)^{5/2} \sqrt{3+5 x}}{20480}-\frac{8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac{783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac{47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac{1042173 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{40960}\\ &=\frac{1042173 (1-2 x)^{3/2} \sqrt{3+5 x}}{409600}-\frac{94743 (1-2 x)^{5/2} \sqrt{3+5 x}}{20480}-\frac{8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac{783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac{47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac{34391709 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{819200}\\ &=\frac{34391709 \sqrt{1-2 x} \sqrt{3+5 x}}{4096000}+\frac{1042173 (1-2 x)^{3/2} \sqrt{3+5 x}}{409600}-\frac{94743 (1-2 x)^{5/2} \sqrt{3+5 x}}{20480}-\frac{8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac{783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac{47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac{378308799 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{8192000}\\ &=\frac{34391709 \sqrt{1-2 x} \sqrt{3+5 x}}{4096000}+\frac{1042173 (1-2 x)^{3/2} \sqrt{3+5 x}}{409600}-\frac{94743 (1-2 x)^{5/2} \sqrt{3+5 x}}{20480}-\frac{8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac{783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac{47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac{378308799 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{4096000 \sqrt{5}}\\ &=\frac{34391709 \sqrt{1-2 x} \sqrt{3+5 x}}{4096000}+\frac{1042173 (1-2 x)^{3/2} \sqrt{3+5 x}}{409600}-\frac{94743 (1-2 x)^{5/2} \sqrt{3+5 x}}{20480}-\frac{8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac{783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac{47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac{3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac{378308799 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{4096000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0594955, size = 80, normalized size = 0.43 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (1843200000 x^6+4387840000 x^5+2867456000 x^4-887043200 x^3-1789716960 x^2-549624420 x+247243887\right )-2648161593 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{286720000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(247243887 - 549624420*x - 1789716960*x^2 - 887043200*x^3 + 2867456000*x^4 +
4387840000*x^5 + 1843200000*x^6) - 2648161593*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/286720000

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Maple [A]  time = 0.009, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{573440000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -36864000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}-87756800000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-57349120000\,{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}-4944877740\,\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)^2*(3+5*x)^(5/2),x)

[Out]

1/573440000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-36864000000*(-10*x^2-x+3)^(1/2)*x^6-87756800000*x^5*(-10*x^2-x+3)^(1
/2)-57349120000*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)-4944877740*(-10*x^2-x+3)^(1/2))/(
-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.52775, size = 157, normalized size = 0.84 \begin{align*} -\frac{9}{14} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} - \frac{157}{112} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{12309}{11200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{8613}{2560} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{8613}{51200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3126519}{204800} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{378308799}{81920000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{3126519}{4096000} \, \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)^2*(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

-9/14*(-10*x^2 - x + 3)^(5/2)*x^2 - 157/112*(-10*x^2 - x + 3)^(5/2)*x - 12309/11200*(-10*x^2 - x + 3)^(5/2) +
8613/2560*(-10*x^2 - x + 3)^(3/2)*x + 8613/51200*(-10*x^2 - x + 3)^(3/2) + 3126519/204800*sqrt(-10*x^2 - x + 3
)*x - 378308799/81920000*sqrt(10)*arcsin(-20/11*x - 1/11) + 3126519/4096000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.53251, size = 352, normalized size = 1.88 \begin{align*} -\frac{1}{28672000} \,{\left (1843200000 \, x^{6} + 4387840000 \, x^{5} + 2867456000 \, x^{4} - 887043200 \, x^{3} - 1789716960 \, x^{2} - 549624420 \, x + 247243887\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{378308799}{81920000} \, \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)^2*(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/28672000*(1843200000*x^6 + 4387840000*x^5 + 2867456000*x^4 - 887043200*x^3 - 1789716960*x^2 - 549624420*x +
247243887)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 378308799/81920000*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)**2*(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.33647, size = 548, normalized size = 2.93 \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)^2*(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-3/7168000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x + 3) - 3968469)*(5*x + 3) + 33
617829)*(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))) - 61/512000000*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)*arcsin(
1/11*sqrt(22)*sqrt(5*x + 3))) - 1/375000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 13640
5)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 17
/384000*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))) + 13/2000*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))) + 9/100*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)))