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

Optimal. Leaf size=137 \[ \frac {3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {239 (1-2 x)^{5/2}}{28 (3 x+2) (5 x+3)^{3/2}}-\frac {13145 (1-2 x)^{3/2}}{84 (5 x+3)^{3/2}}+\frac {13145 \sqrt {1-2 x}}{4 \sqrt {5 x+3}}-\frac {13145}{4} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, 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)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {239 (1-2 x)^{5/2}}{28 (3 x+2) (5 x+3)^{3/2}}-\frac {13145 (1-2 x)^{3/2}}{84 (5 x+3)^{3/2}}+\frac {13145 \sqrt {1-2 x}}{4 \sqrt {5 x+3}}-\frac {13145}{4} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13145*(1 - 2*x)^(3/2))/(84*(3 + 5*x)^(3/2)) + (3*(1 - 2*x)^(7/2))/(14*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (239*(1
- 2*x)^(5/2))/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (13145*Sqrt[1 - 2*x])/(4*Sqrt[3 + 5*x]) - (13145*Sqrt[7]*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4

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)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx &=\frac {3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {239}{28} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {13145}{56} \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {13145 (1-2 x)^{3/2}}{84 (3+5 x)^{3/2}}+\frac {3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}-\frac {13145}{8} \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {13145 (1-2 x)^{3/2}}{84 (3+5 x)^{3/2}}+\frac {3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {13145 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}+\frac {92015}{8} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {13145 (1-2 x)^{3/2}}{84 (3+5 x)^{3/2}}+\frac {3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {13145 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}+\frac {92015}{4} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {13145 (1-2 x)^{3/2}}{84 (3+5 x)^{3/2}}+\frac {3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {13145 \sqrt {1-2 x}}{4 \sqrt {3+5 x}}-\frac {13145}{4} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 78, normalized size = 0.57 \begin {gather*} \frac {1}{12} \left (\frac {\sqrt {1-2 x} \left (1809585 x^3+3458634 x^2+2200321 x+465916\right )}{(3 x+2)^2 (5 x+3)^{3/2}}-39435 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[1 - 2*x]*(465916 + 2200321*x + 3458634*x^2 + 1809585*x^3))/((2 + 3*x)^2*(3 + 5*x)^(3/2)) - 39435*Sqrt[7
]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/12

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IntegrateAlgebraic [A]  time = 2.79, size = 200, normalized size = 1.46 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (361917 \sqrt {5} (5 x+3)^3+201381 \sqrt {5} (5 x+3)^2+21560 \sqrt {5} (5 x+3)-968 \sqrt {5}\right )}{60 (5 x+3)^{3/2} (3 (5 x+3)+1)^2}-\frac {13145}{4} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )-\frac {13145}{4} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-968*Sqrt[5] + 21560*Sqrt[5]*(3 + 5*x) + 201381*Sqrt[5]*(3 + 5*x)^2 + 361917*Sqrt[5]*
(3 + 5*x)^3))/(60*(3 + 5*x)^(3/2)*(1 + 3*(3 + 5*x))^2) - (13145*Sqrt[7]*ArcTan[(Sqrt[2/(34 + Sqrt[1155])]*Sqrt
[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/4 - (13145*Sqrt[7]*ArcTan[(Sqrt[68 + 2*Sqrt[1155]]*Sqrt[3 + 5
*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/4

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fricas [A]  time = 1.13, size = 116, normalized size = 0.85 \begin {gather*} -\frac {39435 \, \sqrt {7} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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 ) - 2 \, {\left (1809585 \, x^{3} + 3458634 \, x^{2} + 2200321 \, x + 465916\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{24 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(39435*sqrt(7)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*
sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 2*(1809585*x^3 + 3458634*x^2 + 2200321*x + 465916)*sqrt(5*x + 3)*sqrt(-2*x
+ 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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giac [B]  time = 2.97, size = 373, normalized size = 2.72 \begin {gather*} \frac {2629}{16} \, \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}{240} \, \sqrt {10} {\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 )}^{3} - \frac {2472 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {9888 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {77 \, {\left (437 \, \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 )}^{3} + 103880 \, \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 )}\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)^(5/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

2629/16*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)))) - 11/240*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)))^3 - 2472*(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) + 9888*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 77/2*(437*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)))^3 + 103880
*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) - sqr
t(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))^2 + 280)^2

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maple [B]  time = 0.02, size = 250, normalized size = 1.82 \begin {gather*} \frac {\left (8872875 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3619170 \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 )+6917268 \sqrt {-10 x^{2}-x +3}\, x^{2}+8991180 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4400642 \sqrt {-10 x^{2}-x +3}\, x +1419660 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+931832 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{24 \left (3 x +2\right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(8872875*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]*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
+3619170*(-10*x^2-x+3)^(1/2)*x^3+8991180*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+6917268*
(-10*x^2-x+3)^(1/2)*x^2+1419660*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4400642*(-10*x^2-x+
3)^(1/2)*x+931832*(-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)^(3/2)

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maxima [A]  time = 1.34, size = 172, normalized size = 1.26 \begin {gather*} \frac {13145}{8} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {40213 \, x}{6 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {69977}{20 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {454757 \, x}{270 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{162 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {25039}{108 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {1473541}{1620 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13145/8*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 40213/6*x/sqrt(-10*x^2 - x + 3) + 69977/20
/sqrt(-10*x^2 - x + 3) + 454757/270*x/(-10*x^2 - x + 3)^(3/2) + 2401/162/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(
-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 25039/108/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 -
x + 3)^(3/2)) - 1473541/1620/(-10*x^2 - x + 3)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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