3.2369 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=151 \[ \frac {4148797 \sqrt {1-2 x} \sqrt {5 x+3}}{28224 (3 x+2)}+\frac {39667 \sqrt {1-2 x} \sqrt {5 x+3}}{2016 (3 x+2)^2}+\frac {227 \sqrt {1-2 x} \sqrt {5 x+3}}{72 (3 x+2)^3}+\frac {7 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}-\frac {5274027 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}} \]

[Out]

-5274027/21952*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+7/12*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x
)^4+227/72*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+39667/2016*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+4148797/2822
4*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.05, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 151, 12, 93, 204} \[ \frac {4148797 \sqrt {1-2 x} \sqrt {5 x+3}}{28224 (3 x+2)}+\frac {39667 \sqrt {1-2 x} \sqrt {5 x+3}}{2016 (3 x+2)^2}+\frac {227 \sqrt {1-2 x} \sqrt {5 x+3}}{72 (3 x+2)^3}+\frac {7 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}-\frac {5274027 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (227*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(72*(2 + 3*x)^3) + (39667
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2016*(2 + 3*x)^2) + (4148797*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28224*(2 + 3*x)) - (
5274027*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 98

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

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)^5 \sqrt {3+5 x}} \, dx &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {\frac {271}{2}-194 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {227 \sqrt {1-2 x} \sqrt {3+5 x}}{72 (2+3 x)^3}+\frac {1}{252} \int \frac {\frac {50183}{4}-15890 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {227 \sqrt {1-2 x} \sqrt {3+5 x}}{72 (2+3 x)^3}+\frac {39667 \sqrt {1-2 x} \sqrt {3+5 x}}{2016 (2+3 x)^2}+\frac {\int \frac {\frac {5978273}{8}-\frac {1388345 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{3528}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {227 \sqrt {1-2 x} \sqrt {3+5 x}}{72 (2+3 x)^3}+\frac {39667 \sqrt {1-2 x} \sqrt {3+5 x}}{2016 (2+3 x)^2}+\frac {4148797 \sqrt {1-2 x} \sqrt {3+5 x}}{28224 (2+3 x)}+\frac {\int \frac {332263701}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{24696}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {227 \sqrt {1-2 x} \sqrt {3+5 x}}{72 (2+3 x)^3}+\frac {39667 \sqrt {1-2 x} \sqrt {3+5 x}}{2016 (2+3 x)^2}+\frac {4148797 \sqrt {1-2 x} \sqrt {3+5 x}}{28224 (2+3 x)}+\frac {5274027 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6272}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {227 \sqrt {1-2 x} \sqrt {3+5 x}}{72 (2+3 x)^3}+\frac {39667 \sqrt {1-2 x} \sqrt {3+5 x}}{2016 (2+3 x)^2}+\frac {4148797 \sqrt {1-2 x} \sqrt {3+5 x}}{28224 (2+3 x)}+\frac {5274027 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{3136}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {227 \sqrt {1-2 x} \sqrt {3+5 x}}{72 (2+3 x)^3}+\frac {39667 \sqrt {1-2 x} \sqrt {3+5 x}}{2016 (2+3 x)^2}+\frac {4148797 \sqrt {1-2 x} \sqrt {3+5 x}}{28224 (2+3 x)}-\frac {5274027 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.52 \[ \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (12446391 x^3+25448120 x^2+17365300 x+3956240\right )}{(3 x+2)^4}-5274027 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3956240 + 17365300*x + 25448120*x^2 + 12446391*x^3))/(2 + 3*x)^4 - 5274027*Sq
rt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/21952

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fricas [A]  time = 0.97, size = 116, normalized size = 0.77 \[ -\frac {5274027 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (12446391 \, x^{3} + 25448120 \, x^{2} + 17365300 \, x + 3956240\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43904 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/43904*(5274027*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(12446391*x^3 + 25448120*x^2 + 17365300*x + 3956240)*sqrt(5*x + 3)*sq
rt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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giac [B]  time = 2.34, size = 368, normalized size = 2.44 \[ \frac {5274027}{439040} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {121 \, \sqrt {10} {\left (113213 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 59365880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 12529809600 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {956821824000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {3827287296000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1568 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5274027/439040*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)))) + 121/1568*sqrt(10)*(113213*((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 + 59365880*((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 + 1252980
9600*((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 + 956821824000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 3827287296000*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)*s
qrt(-10*x + 5) - sqrt(22)))^2 + 280)^4

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maple [B]  time = 0.01, size = 250, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (427196187 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{43904 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/43904*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(427196187*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))[phone]*(-10*x^2-x+3)^(1/2)*x^3+506306592*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+84384432*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+55387360*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3
*x+2)^4

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maxima [A]  time = 1.31, size = 143, normalized size = 0.95 \[ \frac {5274027}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{12 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {227 \, \sqrt {-10 \, x^{2} - x + 3}}{72 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {39667 \, \sqrt {-10 \, x^{2} - x + 3}}{2016 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {4148797 \, \sqrt {-10 \, x^{2} - x + 3}}{28224 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5274027/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 7/12*sqrt(-10*x^2 - x + 3)/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16) + 227/72*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 39667/2016*sqrt(
-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 4148797/28224*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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