3.2413 \(\int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=182 \[ -\frac{3}{70} (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{37}{240} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{407}{960} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{4477 \sqrt{5 x+3} (1-2 x)^{7/2}}{5120}+\frac{49247 \sqrt{5 x+3} (1-2 x)^{5/2}}{153600}+\frac{541717 \sqrt{5 x+3} (1-2 x)^{3/2}}{614400}+\frac{5958887 \sqrt{5 x+3} \sqrt{1-2 x}}{2048000}+\frac{65547757 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048000 \sqrt{10}} \]

[Out]

(5958887*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2048000 + (541717*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/614400 + (49247*(1 - 2*
x)^(5/2)*Sqrt[3 + 5*x])/153600 - (4477*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/5120 - (407*(1 - 2*x)^(7/2)*(3 + 5*x)^(3
/2))/960 - (37*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/240 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))/70 + (65547757*ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2048000*Sqrt[10])

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Rubi [A]  time = 0.0593908, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{3}{70} (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{37}{240} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{407}{960} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{4477 \sqrt{5 x+3} (1-2 x)^{7/2}}{5120}+\frac{49247 \sqrt{5 x+3} (1-2 x)^{5/2}}{153600}+\frac{541717 \sqrt{5 x+3} (1-2 x)^{3/2}}{614400}+\frac{5958887 \sqrt{5 x+3} \sqrt{1-2 x}}{2048000}+\frac{65547757 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5958887*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2048000 + (541717*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/614400 + (49247*(1 - 2*
x)^(5/2)*Sqrt[3 + 5*x])/153600 - (4477*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/5120 - (407*(1 - 2*x)^(7/2)*(3 + 5*x)^(3
/2))/960 - (37*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/240 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))/70 + (65547757*ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2048000*Sqrt[10])

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)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx &=-\frac{3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac{37}{20} \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx\\ &=-\frac{37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac{3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac{407}{96} \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac{3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac{4477}{640} \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx\\ &=-\frac{4477 (1-2 x)^{7/2} \sqrt{3+5 x}}{5120}-\frac{407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac{3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac{49247 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{10240}\\ &=\frac{49247 (1-2 x)^{5/2} \sqrt{3+5 x}}{153600}-\frac{4477 (1-2 x)^{7/2} \sqrt{3+5 x}}{5120}-\frac{407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac{3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac{541717 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{61440}\\ &=\frac{541717 (1-2 x)^{3/2} \sqrt{3+5 x}}{614400}+\frac{49247 (1-2 x)^{5/2} \sqrt{3+5 x}}{153600}-\frac{4477 (1-2 x)^{7/2} \sqrt{3+5 x}}{5120}-\frac{407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac{3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac{5958887 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{409600}\\ &=\frac{5958887 \sqrt{1-2 x} \sqrt{3+5 x}}{2048000}+\frac{541717 (1-2 x)^{3/2} \sqrt{3+5 x}}{614400}+\frac{49247 (1-2 x)^{5/2} \sqrt{3+5 x}}{153600}-\frac{4477 (1-2 x)^{7/2} \sqrt{3+5 x}}{5120}-\frac{407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac{3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac{65547757 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{4096000}\\ &=\frac{5958887 \sqrt{1-2 x} \sqrt{3+5 x}}{2048000}+\frac{541717 (1-2 x)^{3/2} \sqrt{3+5 x}}{614400}+\frac{49247 (1-2 x)^{5/2} \sqrt{3+5 x}}{153600}-\frac{4477 (1-2 x)^{7/2} \sqrt{3+5 x}}{5120}-\frac{407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac{3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac{65547757 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{2048000 \sqrt{5}}\\ &=\frac{5958887 \sqrt{1-2 x} \sqrt{3+5 x}}{2048000}+\frac{541717 (1-2 x)^{3/2} \sqrt{3+5 x}}{614400}+\frac{49247 (1-2 x)^{5/2} \sqrt{3+5 x}}{153600}-\frac{4477 (1-2 x)^{7/2} \sqrt{3+5 x}}{5120}-\frac{407}{960} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{37}{240} (1-2 x)^{7/2} (3+5 x)^{5/2}-\frac{3}{70} (1-2 x)^{7/2} (3+5 x)^{7/2}+\frac{65547757 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{2048000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.075419, size = 80, normalized size = 0.44 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (1843200000 x^6+1879040000 x^5-1272064000 x^4-1600483200 x^3+287177440 x^2+540576580 x-24901623\right )-1376502897 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{430080000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-24901623 + 540576580*x + 287177440*x^2 - 1600483200*x^3 - 1272064000*x^4 + 1
879040000*x^5 + 1843200000*x^6) - 1376502897*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/430080000

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Maple [A]  time = 0.009, size = 155, normalized size = 0.9 \begin{align*}{\frac{1}{860160000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 36864000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-25441280000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-32009664000\,{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}-498032460\,\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)^(5/2)*(2+3*x)*(3+5*x)^(5/2),x)

[Out]

1/860160000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(36864000000*(-10*x^2-x+3)^(1/2)*x^6+37580800000*x^5*(-10*x^2-x+3)^(1/
2)-25441280000*x^4*(-10*x^2-x+3)^(1/2)-32009664000*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)+
1376502897*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-498032460*(-10*x^2-x+3)^(1/2))/(-10
*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.13296, size = 153, normalized size = 0.84 \begin{align*} -\frac{3}{70} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{37}{120} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{37}{2400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{4477}{3840} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{4477}{76800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{541717}{102400} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{65547757}{40960000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{541717}{2048000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/70*(-10*x^2 - x + 3)^(7/2) + 37/120*(-10*x^2 - x + 3)^(5/2)*x + 37/2400*(-10*x^2 - x + 3)^(5/2) + 4477/3840
*(-10*x^2 - x + 3)^(3/2)*x + 4477/76800*(-10*x^2 - x + 3)^(3/2) + 541717/102400*sqrt(-10*x^2 - x + 3)*x - 6554
7757/40960000*sqrt(10)*arcsin(-20/11*x - 1/11) + 541717/2048000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.73654, size = 348, normalized size = 1.91 \begin{align*} \frac{1}{43008000} \,{\left (1843200000 \, x^{6} + 1879040000 \, x^{5} - 1272064000 \, x^{4} - 1600483200 \, x^{3} + 287177440 \, x^{2} + 540576580 \, x - 24901623\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{65547757}{40960000} \, \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)^(5/2)*(2+3*x)*(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

1/43008000*(1843200000*x^6 + 1879040000*x^5 - 1272064000*x^4 - 1600483200*x^3 + 287177440*x^2 + 540576580*x -
24901623)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 65547757/40960000*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)**(5/2)*(2+3*x)*(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.99223, size = 548, normalized size = 3.01 \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)^(5/2)*(2+3*x)*(3+5*x)^(5/2),x, algorithm="giac")

[Out]

1/3584000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x + 3) - 3968469)*(5*x + 3) + 336
17829)*(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))) + 13/384000000*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))) - 137/192000000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 1
36405)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
- 17/240000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4
5375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/1600*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))) + 9/200*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)))