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

Optimal. Leaf size=151 \[ \frac {57595 \sqrt {1-2 x} \sqrt {5 x+3}}{197568 (3 x+2)}+\frac {85 \sqrt {1-2 x} \sqrt {5 x+3}}{14112 (3 x+2)^2}-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{504 (3 x+2)^3}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}-\frac {78045 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \]

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Rubi [A]  time = 0.05, 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} \begin {gather*} \frac {57595 \sqrt {1-2 x} \sqrt {5 x+3}}{197568 (3 x+2)}+\frac {85 \sqrt {1-2 x} \sqrt {5 x+3}}{14112 (3 x+2)^2}-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{504 (3 x+2)^3}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}-\frac {78045 \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[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^5),x]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^4) - (43*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(504*(2 + 3*x)^3) + (85*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) + (57595*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (78045
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*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 {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {1}{84} \int \frac {-\frac {793}{2}-670 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}-\frac {\int \frac {-\frac {8225}{4}-3010 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{1764}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}-\frac {\int \frac {-\frac {126455}{8}+\frac {2975 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{24696}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}-\frac {\int -\frac {4916835}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{172872}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}+\frac {78045 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{43904}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}+\frac {78045 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{21952}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}-\frac {78045 \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 (172785 x^3+346760 x^2+226348 x+48240\right )}{(3 x+2)^4}-78045 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(48240 + 226348*x + 346760*x^2 + 172785*x^3))/(2 + 3*x)^4 - 78045*Sqrt[7]*ArcT
an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/153664

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IntegrateAlgebraic [A]  time = 0.30, size = 122, normalized size = 0.81 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {645 (1-2 x)^3}{(5 x+3)^3}+\frac {16555 (1-2 x)^2}{(5 x+3)^2}-\frac {65709 (1-2 x)}{5 x+3}-196147\right )}{21952 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^4}-\frac {78045 \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[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^5),x]

[Out]

(-121*Sqrt[1 - 2*x]*(-196147 + (645*(1 - 2*x)^3)/(3 + 5*x)^3 + (16555*(1 - 2*x)^2)/(3 + 5*x)^2 - (65709*(1 - 2
*x))/(3 + 5*x)))/(21952*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^4) - (78045*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt
[3 + 5*x])])/(21952*Sqrt[7])

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fricas [A]  time = 1.36, size = 116, normalized size = 0.77 \begin {gather*} -\frac {78045 \, \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 (172785 \, x^{3} + 346760 \, x^{2} + 226348 \, x + 48240\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\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)^(3/2)/(2+3*x)^5/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/307328*(78045*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*(172785*x^3 + 346760*x^2 + 226348*x + 48240)*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 = 2.58, size = 368, normalized size = 2.44 \begin {gather*} \frac {15609}{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 {605 \, \sqrt {10} {\left (129 \, {\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} + 132440 \, {\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} - 21026880 \, {\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 {2510681600 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {10042726400 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{10976 \, {\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)^(3/2)/(2+3*x)^5/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

15609/614656*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)))) - 605/10976*sqrt(10)*(129*((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 + 132440*((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 - 21026880*((sq
rt(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 - 25
10681600*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 10042726400*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)^4

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maple [B]  time = 0.02, size = 250, normalized size = 1.66 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (6321645 \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 )+2418990 \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 )+4854640 \sqrt {-10 x^{2}-x +3}\, x^{2}+7492320 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3168872 \sqrt {-10 x^{2}-x +3}\, x +1248720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+675360 \sqrt {-10 x^{2}-x +3}\right )}{307328 \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)^(3/2)/(3*x+2)^5/(-2*x+1)^(1/2),x)

[Out]

1/307328*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(6321645*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+
16857720*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))+2418990*(-10*x^2-x+3)^(1/2)*x^3+7492320*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))+4854640*(-10*x^2-x+3)^(1/2)*x^2+1248720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^
2-x+3)^(1/2))+3168872*(-10*x^2-x+3)^(1/2)*x+675360*(-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 = 1.32, size = 143, normalized size = 0.95 \begin {gather*} \frac {78045}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{84 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {43 \, \sqrt {-10 \, x^{2} - x + 3}}{504 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {85 \, \sqrt {-10 \, x^{2} - x + 3}}{14112 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {57595 \, \sqrt {-10 \, x^{2} - x + 3}}{197568 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

78045/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/84*sqrt(-10*x^2 - x + 3)/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16) - 43/504*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 85/14112*sqrt(-10
*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 57595/197568*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 {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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