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

Optimal. Leaf size=115 \[ \frac {3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt {5 x+3}}+\frac {173 (1-2 x)^{3/2}}{28 (3 x+2) \sqrt {5 x+3}}-\frac {5709 \sqrt {1-2 x}}{28 \sqrt {5 x+3}}+\frac {5709 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4 \sqrt {7}} \]

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Rubi [A]  time = 0.03, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \begin {gather*} \frac {3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt {5 x+3}}+\frac {173 (1-2 x)^{3/2}}{28 (3 x+2) \sqrt {5 x+3}}-\frac {5709 \sqrt {1-2 x}}{28 \sqrt {5 x+3}}+\frac {5709 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5709*Sqrt[1 - 2*x])/(28*Sqrt[3 + 5*x]) + (3*(1 - 2*x)^(5/2))/(14*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (173*(1 - 2*x)
^(3/2))/(28*(2 + 3*x)*Sqrt[3 + 5*x]) + (5709*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4*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 96

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))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[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)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx &=\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173}{28} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 (1-2 x)^{3/2}}{28 (2+3 x) \sqrt {3+5 x}}+\frac {5709}{56} \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {5709 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 (1-2 x)^{3/2}}{28 (2+3 x) \sqrt {3+5 x}}-\frac {5709}{8} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {5709 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 (1-2 x)^{3/2}}{28 (2+3 x) \sqrt {3+5 x}}-\frac {5709}{4} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {5709 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {3 (1-2 x)^{5/2}}{14 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 (1-2 x)^{3/2}}{28 (2+3 x) \sqrt {3+5 x}}+\frac {5709 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 74, normalized size = 0.64 \begin {gather*} \frac {5709 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4 \sqrt {7}}-\frac {\sqrt {1-2 x} \left (7485 x^2+9815 x+3212\right )}{4 (3 x+2)^2 \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[1 - 2*x]*(3212 + 9815*x + 7485*x^2))/((2 + 3*x)^2*Sqrt[3 + 5*x]) + (5709*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(4*Sqrt[7])

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IntegrateAlgebraic [A]  time = 2.73, size = 186, normalized size = 1.62 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (-1497 \sqrt {5} (5 x+3)^2-833 \sqrt {5} (5 x+3)-88 \sqrt {5}\right )}{4 \sqrt {5 x+3} (3 (5 x+3)+1)^2}+\frac {5709 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{4 \sqrt {7}}+\frac {5709 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{4 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-88*Sqrt[5] - 833*Sqrt[5]*(3 + 5*x) - 1497*Sqrt[5]*(3 + 5*x)^2))/(4*Sqrt[3 + 5*x]*(1
+ 3*(3 + 5*x))^2) + (5709*ArcTan[(Sqrt[2/(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])
])/(4*Sqrt[7]) + (5709*ArcTan[(Sqrt[68 + 2*Sqrt[1155]]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(4
*Sqrt[7])

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fricas [A]  time = 1.52, size = 101, normalized size = 0.88 \begin {gather*} \frac {5709 \, \sqrt {7} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\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 (7485 \, x^{2} + 9815 \, x + 3212\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{56 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end {gather*}

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]

1/56*(5709*sqrt(7)*(45*x^3 + 87*x^2 + 56*x + 12)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/
(10*x^2 + x - 3)) - 14*(7485*x^2 + 9815*x + 3212)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x + 12)

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giac [B]  time = 1.74, size = 311, normalized size = 2.70 \begin {gather*} -\frac {5709}{560} \, \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 {11}{2} \, \sqrt {10} {\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 )} - \frac {55 \, \sqrt {10} {\left (61 \, {\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 {13384 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {53536 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{2 \, {\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 )}^{2}} \end {gather*}

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]

-5709/560*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)))) - 11/2*sqrt(10)*((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))) - 55/2*sqrt(10)*(61*((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 + 13384*(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 53536*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((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)))^2 + 280)^2

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maple [B]  time = 0.02, size = 202, normalized size = 1.76 \begin {gather*} -\frac {\left (256905 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+496683 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+104790 \sqrt {-10 x^{2}-x +3}\, x^{2}+319704 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+137410 \sqrt {-10 x^{2}-x +3}\, x +68508 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+44968 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{56 \left (3 x +2\right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56*(256905*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+496683*7^(1/2)*x^2*arctan(1/14*(3
7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+319704*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+10479
0*(-10*x^2-x+3)^(1/2)*x^2+68508*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+137410*(-10*x^2-x+3
)^(1/2)*x+44968*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(3*x+2)^2/(-10*x^2-x+3)^(1/2)/(5*x+3)^(1/2)

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maxima [A]  time = 1.28, size = 143, normalized size = 1.24 \begin {gather*} -\frac {5709}{56} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2495 \, x}{6 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2605}{12 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{18 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {1127}{36 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}

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]

-5709/56*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2495/6*x/sqrt(-10*x^2 - x + 3) - 2605/12/
sqrt(-10*x^2 - x + 3) + 49/18/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x +
3)) + 1127/36/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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