3.2408 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac {121 \sqrt {1-2 x} (5 x+3)^{5/2}}{16 (3 x+2)^3}+\frac {11 (1-2 x)^{3/2} (5 x+3)^{5/2}}{8 (3 x+2)^4}+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}-\frac {1331 \sqrt {1-2 x} (5 x+3)^{3/2}}{448 (3 x+2)^2}-\frac {43923 \sqrt {1-2 x} \sqrt {5 x+3}}{6272 (3 x+2)}-\frac {483153 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \]

[Out]

1/5*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^5+11/8*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4-483153/43904*arctan(1/7*(
1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1331/448*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2+121/16*(3+5*x)^(5/2
)*(1-2*x)^(1/2)/(2+3*x)^3-43923/6272*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.05, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac {121 \sqrt {1-2 x} (5 x+3)^{5/2}}{16 (3 x+2)^3}+\frac {11 (1-2 x)^{3/2} (5 x+3)^{5/2}}{8 (3 x+2)^4}+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}-\frac {1331 \sqrt {1-2 x} (5 x+3)^{3/2}}{448 (3 x+2)^2}-\frac {43923 \sqrt {1-2 x} \sqrt {5 x+3}}{6272 (3 x+2)}-\frac {483153 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]

[Out]

(-43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6272*(2 + 3*x)) - (1331*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(448*(2 + 3*x)^2)
+ ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(5*(2 + 3*x)^5) + (11*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(8*(2 + 3*x)^4) +
(121*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(16*(2 + 3*x)^3) - (483153*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
(6272*Sqrt[7])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 94

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx &=\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac {11}{2} \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac {363}{16} \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}+\frac {1331}{32} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}+\frac {43923}{896} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}+\frac {483153 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{12544}\\ &=-\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}+\frac {483153 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{6272}\\ &=-\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}-\frac {1331 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac {121 \sqrt {1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}-\frac {483153 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 109, normalized size = 0.61 \[ \frac {11 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )}{(3 x+2)^4}-43923 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{43904}+\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]

[Out]

((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(5*(2 + 3*x)^5) + (11*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(32400 + 145940*x + 21
3240*x^2 + 100159*x^3))/(2 + 3*x)^4 - 43923*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/43904

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fricas [A]  time = 1.01, size = 131, normalized size = 0.73 \[ -\frac {2415765 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (17153435 \, x^{4} + 46327530 \, x^{3} + 47166452 \, x^{2} + 21361768 \, x + 3620448\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{439040 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/439040*(2415765*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x + 2
0)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(17153435*x^4 + 46327530*x^3 + 47166452*x^2 + 21361768*
x + 3620448)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [B]  time = 3.36, size = 426, normalized size = 2.37 \[ \frac {483153}{878080} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {161051 \, \sqrt {10} {\left (3 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 3920 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 2007040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 307328000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {18439680000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {73758720000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3136 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

483153/878080*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(
22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 161051/3136*sqrt(10)*(3*((sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 3920*((sqrt(2)*sqrt(-
10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 - 2007040*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 3073
28000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22
)))^3 - 18439680000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 73758720000*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^2 + 280)^5

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maple [B]  time = 0.01, size = 298, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (587030895 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{439040 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)*(5*x+3)^(3/2)/(3*x+2)^6,x)

[Out]

1/439040*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(587030895*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)[phone]*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^3*arctan(1/14
*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^4+1739350800*7^(1/2)*x^2*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3+579783600*7^(1/2)*x*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2+77304480*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+50686272*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*
x+2)^5

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maxima [A]  time = 1.40, size = 227, normalized size = 1.26 \[ \frac {90695}{32928} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{56 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1221 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{784 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {54417 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {738705}{21952} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {483153}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {650859}{43904} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {215303 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{131712 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

90695/32928*(-10*x^2 - x + 3)^(3/2) + 1/5*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 24
0*x + 32) + 33/56*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1221/784*(-10*x^2 - x + 3
)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 54417/21952*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 738705/21952*s
qrt(-10*x^2 - x + 3)*x + 483153/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 650859/43904
*sqrt(-10*x^2 - x + 3) + 215303/131712*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2)^6,x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2)^6, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**6,x)

[Out]

Timed out

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