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

Optimal. Leaf size=104 \[ \frac {3 (1-2 x)^{5/2}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac {169 (1-2 x)^{3/2}}{21 (5 x+3)^{3/2}}+\frac {169 \sqrt {1-2 x}}{\sqrt {5 x+3}}-169 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 104, 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}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac {169 (1-2 x)^{3/2}}{21 (5 x+3)^{3/2}}+\frac {169 \sqrt {1-2 x}}{\sqrt {5 x+3}}-169 \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)^(3/2)/((2 + 3*x)^2*(3 + 5*x)^(5/2)),x]

[Out]

(-169*(1 - 2*x)^(3/2))/(21*(3 + 5*x)^(3/2)) + (3*(1 - 2*x)^(5/2))/(7*(2 + 3*x)*(3 + 5*x)^(3/2)) + (169*Sqrt[1
- 2*x])/Sqrt[3 + 5*x] - 169*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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)^2 (3+5 x)^{5/2}} \, dx &=\frac {3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {169}{14} \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {169 (1-2 x)^{3/2}}{21 (3+5 x)^{3/2}}+\frac {3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}-\frac {169}{2} \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {169 (1-2 x)^{3/2}}{21 (3+5 x)^{3/2}}+\frac {3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {169 \sqrt {1-2 x}}{\sqrt {3+5 x}}+\frac {1183}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {169 (1-2 x)^{3/2}}{21 (3+5 x)^{3/2}}+\frac {3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {169 \sqrt {1-2 x}}{\sqrt {3+5 x}}+1183 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {169 (1-2 x)^{3/2}}{21 (3+5 x)^{3/2}}+\frac {3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {169 \sqrt {1-2 x}}{\sqrt {3+5 x}}-169 \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 = 0.04, size = 92, normalized size = 0.88 \begin {gather*} \frac {\sqrt {1-2 x} \left (7755 x^2+9652 x+2995\right )-507 \sqrt {7} \sqrt {5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 (3 x+2) (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(2995 + 9652*x + 7755*x^2) - 507*Sqrt[7]*Sqrt[3 + 5*x]*(6 + 19*x + 15*x^2)*ArcTan[Sqrt[1 - 2*x]
/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*(2 + 3*x)*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.15, size = 115, normalized size = 1.11 \begin {gather*} \frac {-\frac {10 (1-2 x)^{5/2}}{(5 x+3)^{5/2}}+\frac {338 (1-2 x)^{3/2}}{(5 x+3)^{3/2}}+\frac {3549 \sqrt {1-2 x}}{\sqrt {5 x+3}}}{3 \left (\frac {1-2 x}{5 x+3}+7\right )}-169 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*(1 - 2*x)^(5/2))/(3 + 5*x)^(5/2) + (338*(1 - 2*x)^(3/2))/(3 + 5*x)^(3/2) + (3549*Sqrt[1 - 2*x])/Sqrt[3 +
5*x])/(3*(7 + (1 - 2*x)/(3 + 5*x))) - 169*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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fricas [A]  time = 1.43, size = 101, normalized size = 0.97 \begin {gather*} -\frac {507 \, \sqrt {7} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\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 (7755 \, x^{2} + 9652 \, x + 2995\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end {gather*}

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]

-1/6*(507*sqrt(7)*(75*x^3 + 140*x^2 + 87*x + 18)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/
(10*x^2 + x - 3)) - 2*(7755*x^2 + 9652*x + 2995)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 18)

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giac [B]  time = 1.93, size = 309, normalized size = 2.97 \begin {gather*} \frac {169}{20} \, \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 {1}{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 {1632 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {6528 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {462 \, \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 )}}{{\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} \end {gather*}

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]

169/20*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)))) - 1/240*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 - 1632*(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))/sqrt(5*x + 3) + 6528*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 462*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)))/(((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)

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maple [B]  time = 0.02, size = 202, normalized size = 1.94 \begin {gather*} \frac {\left (38025 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+70980 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+15510 \sqrt {-10 x^{2}-x +3}\, x^{2}+44109 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+19304 \sqrt {-10 x^{2}-x +3}\, x +9126 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+5990 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{6 \left (3 x +2\right ) \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)^(3/2)/(3*x+2)^2/(5*x+3)^(5/2),x)

[Out]

1/6*(38025*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+70980*7^(1/2)*x^2*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+44109*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+15510*(-10
*x^2-x+3)^(1/2)*x^2+9126*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+19304*(-10*x^2-x+3)^(1/2)*
x+5990*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(3*x+2)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(3/2)

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maxima [A]  time = 1.23, size = 121, normalized size = 1.16 \begin {gather*} \frac {169}{2} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {1034 \, x}{3 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {2699}{15 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3902 \, x}{45 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {343}{27 \, {\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 {6343}{135 \, {\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)^(3/2)/(2+3*x)^2/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

169/2*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1034/3*x/sqrt(-10*x^2 - x + 3) + 2699/15/sqr
t(-10*x^2 - x + 3) + 3902/45*x/(-10*x^2 - x + 3)^(3/2) + 343/27/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x
+ 3)^(3/2)) - 6343/135/(-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 )}^{3/2}}{{\left (3\,x+2\right )}^2\,{\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)^(3/2)/((3*x + 2)^2*(5*x + 3)^(5/2)),x)

[Out]

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

[Out]

Timed out

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