3.26.96 \(\int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\) [2596]

Optimal. Leaf size=144 \[ \frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}} \]

[Out]

-5/196*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+11/21*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^2+5/4
2*(3+5*x)^(1/2)/(1-2*x)^(1/2)-3/14*(3+5*x)^(1/2)/(2+3*x)^2/(1-2*x)^(1/2)-5/28*(3+5*x)^(1/2)/(2+3*x)/(1-2*x)^(1
/2)

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Rubi [A]
time = 0.03, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 156, 157, 12, 95, 210} \begin {gather*} -\frac {5 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}}+\frac {5 \sqrt {5 x+3}}{42 \sqrt {1-2 x}}-\frac {5 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)}-\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[3 + 5*x])/(42*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - (3*Sqrt[3 + 5*x])
/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*Sqrt[3 + 5*x])/(28*Sqrt[1 - 2*x]*(2 + 3*x)) - (5*ArcTan[Sqrt[1 - 2*x]/(Sq
rt[7]*Sqrt[3 + 5*x])])/(28*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 95

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 100

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 156

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] && ILtQ[m, -1]

Rule 157

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] && IntegersQ[2*m, 2*n, 2*p]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 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}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {1}{21} \int \frac {-134-\frac {465 x}{2}}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {1}{294} \int \frac {-\frac {1435}{2}-1260 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}-\frac {\int \frac {-\frac {11515}{4}-3675 x}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{2058}\\ &=\frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}+\frac {\int \frac {56595}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{79233}\\ &=\frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}+\frac {5}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}+\frac {5}{28} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 95, normalized size = 0.66 \begin {gather*} -\frac {7 \sqrt {3+5 x} \left (-36-91 x+60 x^2+180 x^3\right )-15 \sqrt {7-14 x} (-1+2 x) (2+3 x)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{588 (1-2 x)^{3/2} (2+3 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/588*(7*Sqrt[3 + 5*x]*(-36 - 91*x + 60*x^2 + 180*x^3) - 15*Sqrt[7 - 14*x]*(-1 + 2*x)*(2 + 3*x)^2*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/((1 - 2*x)^(3/2)*(2 + 3*x)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(111)=222\).
time = 0.08, size = 257, normalized size = 1.78

method	result	size
default	\(\frac {\left (540 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+180 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-345 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-2520 x^{3} \sqrt {-10 x^{2}-x +3}-60 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -840 x^{2} \sqrt {-10 x^{2}-x +3}+60 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1274 x \sqrt {-10 x^{2}-x +3}+504 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1176 \left (2+3 x \right )^{2} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1176*(540*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+180*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-345*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-2520*x^3*(-
10*x^2-x+3)^(1/2)-60*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-840*x^2*(-10*x^2-x+3)^(1/2)+
60*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1274*x*(-10*x^2-x+3)^(1/2)+504*(-10*x^2-x+3)^(1/
2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.56, size = 172, normalized size = 1.19 \begin {gather*} \frac {5}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {25 \, x}{42 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {5}{84 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {125 \, x}{126 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1}{378 \, {\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 {43}{756 \, {\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 {205}{252 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 25/42*x/sqrt(-10*x^2 - x + 3) + 5/84/sqrt(-1
0*x^2 - x + 3) + 125/126*x/(-10*x^2 - x + 3)^(3/2) + 1/378/(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)) - 43/756/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 2
05/252/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]
time = 0.50, size = 116, normalized size = 0.81 \begin {gather*} -\frac {15 \, \sqrt {7} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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 (180 \, x^{3} + 60 \, x^{2} - 91 \, x - 36\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1176 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1176*(15*sqrt(7)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2
*x + 1)/(10*x^2 + x - 3)) + 14*(180*x^3 + 60*x^2 - 91*x - 36)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 -
23*x^2 - 4*x + 4)

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Sympy [F(-1)] Timed out
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)/(1-2*x)**(5/2)/(2+3*x)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (111) = 222\).
time = 1.61, size = 291, normalized size = 2.02 \begin {gather*} \frac {1}{784} \, \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 {8 \, {\left (157 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1056 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{180075 \, {\left (2 \, x - 1\right )}^{2}} - \frac {33 \, \sqrt {10} {\left (83 \, {\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 {41720 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {166880 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4802 \, {\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((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^3,x, algorithm="giac")

[Out]

1/784*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)))) - 8/180075*(157*sqrt(5)*(5*x + 3) - 1056*sqrt(5))*sqrt(5*
x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 33/4802*sqrt(10)*(83*((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 + 41720*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*
x + 3) - 166880*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^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}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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