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

Optimal. Leaf size=151 \[ \frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}-\frac {11 \sqrt {1-2 x} (5 x+3)^{5/2}}{168 (3 x+2)^3}-\frac {605 \sqrt {1-2 x} (5 x+3)^{3/2}}{4704 (3 x+2)^2}-\frac {6655 \sqrt {1-2 x} \sqrt {5 x+3}}{21952 (3 x+2)}-\frac {73205 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \]

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Rubi [A]  time = 0.04, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}-\frac {11 \sqrt {1-2 x} (5 x+3)^{5/2}}{168 (3 x+2)^3}-\frac {605 \sqrt {1-2 x} (5 x+3)^{3/2}}{4704 (3 x+2)^2}-\frac {6655 \sqrt {1-2 x} \sqrt {5 x+3}}{21952 (3 x+2)}-\frac {73205 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6655*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (605*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(4704*(2 + 3*x)^2)
- (11*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(168*(2 + 3*x)^3) + (Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(4*(2 + 3*x)^4) - (7
3205*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*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 {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^{7/2}}{4 (2+3 x)^4}+\frac {11}{8} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac {\sqrt {1-2 x} (3+5 x)^{7/2}}{4 (2+3 x)^4}+\frac {605}{336} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}-\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac {\sqrt {1-2 x} (3+5 x)^{7/2}}{4 (2+3 x)^4}+\frac {6655 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{3136}\\ &=-\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}-\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac {\sqrt {1-2 x} (3+5 x)^{7/2}}{4 (2+3 x)^4}+\frac {73205 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{43904}\\ &=-\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}-\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac {\sqrt {1-2 x} (3+5 x)^{7/2}}{4 (2+3 x)^4}+\frac {73205 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{21952}\\ &=-\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}-\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac {\sqrt {1-2 x} (3+5 x)^{7/2}}{4 (2+3 x)^4}-\frac {73205 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.52 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (814395 x^3+1285720 x^2+654436 x+105552\right )}{(3 x+2)^4}-219615 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{460992} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(105552 + 654436*x + 1285720*x^2 + 814395*x^3))/(2 + 3*x)^4 - 219615*Sqrt[7]*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/460992

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IntegrateAlgebraic [A]  time = 0.26, size = 122, normalized size = 0.81 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {15 (1-2 x)^3}{(5 x+3)^3}+\frac {385 (1-2 x)^2}{(5 x+3)^2}+\frac {3577 (1-2 x)}{5 x+3}-5145\right )}{65856 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^4}-\frac {73205 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*Sqrt[1 - 2*x]*(-5145 + (15*(1 - 2*x)^3)/(3 + 5*x)^3 + (385*(1 - 2*x)^2)/(3 + 5*x)^2 + (3577*(1 - 2*x))
/(3 + 5*x)))/(65856*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^4) - (73205*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(21952*Sqrt[7])

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fricas [A]  time = 1.51, size = 116, normalized size = 0.77 \begin {gather*} -\frac {219615 \, \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 (814395 \, x^{3} + 1285720 \, x^{2} + 654436 \, x + 105552\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{921984 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/921984*(219615*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*(814395*x^3 + 1285720*x^2 + 654436*x + 105552)*sqrt(5*x + 3)*sqrt(-2*
x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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giac [B]  time = 2.65, size = 368, normalized size = 2.44 \begin {gather*} \frac {14641}{614656} \, \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 {73205 \, \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 )}^{7} + 3080 \, {\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} + 1144640 \, {\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 {65856000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {263424000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{32928 \, {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14641/614656*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 73205/32928*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)))^7 + 3080*((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)))^5 + 1144640*((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 - 65856
000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 263424000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(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)^4

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maple [B]  time = 0.01, size = 250, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (17788815 \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 )+9162104 \sqrt {-10 x^{2}-x +3}\, x +3513840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1477728 \sqrt {-10 x^{2}-x +3}\right )}{921984 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/921984*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(17788815*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+21083040*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+3513840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))+9162104*(-10*x^2-x+3)^(1/2)*x+1477728*(-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 = 1.26, size = 157, normalized size = 1.04 \begin {gather*} \frac {73205}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {3025}{16464} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{84 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {125 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1176 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1815 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {22385 \, \sqrt {-10 \, x^{2} - x + 3}}{65856 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

73205/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 3025/16464*sqrt(-10*x^2 - x + 3) + 1/
84*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) - 125/1176*(-10*x^2 - x + 3)^(3/2)/(27*x^3
+ 54*x^2 + 36*x + 8) + 1815/10976*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 22385/65856*sqrt(-10*x^2 - x +
3)/(3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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