3.25.10 \(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx\)

Optimal. Leaf size=209 \[ \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}+\frac {426781 \sqrt {1-2 x} \sqrt {5 x+3}}{6453888 (3 x+2)}-\frac {55277 \sqrt {1-2 x} \sqrt {5 x+3}}{460992 (3 x+2)^2}-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{82320 (3 x+2)^3}-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{41160 (3 x+2)^4}+\frac {164 \sqrt {1-2 x} \sqrt {5 x+3}}{735 (3 x+2)^5}-\frac {3474273 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2151296 \sqrt {7}} \]

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Rubi [A]  time = 0.08, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 149, 151, 12, 93, 204} \begin {gather*} \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}+\frac {426781 \sqrt {1-2 x} \sqrt {5 x+3}}{6453888 (3 x+2)}-\frac {55277 \sqrt {1-2 x} \sqrt {5 x+3}}{460992 (3 x+2)^2}-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{82320 (3 x+2)^3}-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{41160 (3 x+2)^4}+\frac {164 \sqrt {1-2 x} \sqrt {5 x+3}}{735 (3 x+2)^5}-\frac {3474273 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2151296 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

(164*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^5) - (42863*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41160*(2 + 3*x)^4)
- (29297*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(82320*(2 + 3*x)^3) - (55277*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(460992*(2 + 3
*x)^2) + (426781*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6453888*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 +
3*x)^5) - (3474273*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2151296*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 149

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*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx &=\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}-\frac {1}{7} \int \frac {\left (-327-\frac {1145 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^6} \, dx\\ &=\frac {164 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^5}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}-\frac {1}{735} \int \frac {-\frac {110549}{2}-\frac {187255 x}{2}}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=\frac {164 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^5}-\frac {42863 \sqrt {1-2 x} \sqrt {3+5 x}}{41160 (2+3 x)^4}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}-\frac {\int \frac {-\frac {1509441}{4}-642945 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{20580}\\ &=\frac {164 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^5}-\frac {42863 \sqrt {1-2 x} \sqrt {3+5 x}}{41160 (2+3 x)^4}-\frac {29297 \sqrt {1-2 x} \sqrt {3+5 x}}{82320 (2+3 x)^3}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}-\frac {\int \frac {-\frac {14471625}{8}-3076185 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{432180}\\ &=\frac {164 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^5}-\frac {42863 \sqrt {1-2 x} \sqrt {3+5 x}}{41160 (2+3 x)^4}-\frac {29297 \sqrt {1-2 x} \sqrt {3+5 x}}{82320 (2+3 x)^3}-\frac {55277 \sqrt {1-2 x} \sqrt {3+5 x}}{460992 (2+3 x)^2}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}-\frac {\int \frac {-\frac {92325135}{16}-\frac {29020425 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{6050520}\\ &=\frac {164 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^5}-\frac {42863 \sqrt {1-2 x} \sqrt {3+5 x}}{41160 (2+3 x)^4}-\frac {29297 \sqrt {1-2 x} \sqrt {3+5 x}}{82320 (2+3 x)^3}-\frac {55277 \sqrt {1-2 x} \sqrt {3+5 x}}{460992 (2+3 x)^2}+\frac {426781 \sqrt {1-2 x} \sqrt {3+5 x}}{6453888 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}-\frac {\int -\frac {1094395995}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{42353640}\\ &=\frac {164 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^5}-\frac {42863 \sqrt {1-2 x} \sqrt {3+5 x}}{41160 (2+3 x)^4}-\frac {29297 \sqrt {1-2 x} \sqrt {3+5 x}}{82320 (2+3 x)^3}-\frac {55277 \sqrt {1-2 x} \sqrt {3+5 x}}{460992 (2+3 x)^2}+\frac {426781 \sqrt {1-2 x} \sqrt {3+5 x}}{6453888 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}+\frac {3474273 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{4302592}\\ &=\frac {164 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^5}-\frac {42863 \sqrt {1-2 x} \sqrt {3+5 x}}{41160 (2+3 x)^4}-\frac {29297 \sqrt {1-2 x} \sqrt {3+5 x}}{82320 (2+3 x)^3}-\frac {55277 \sqrt {1-2 x} \sqrt {3+5 x}}{460992 (2+3 x)^2}+\frac {426781 \sqrt {1-2 x} \sqrt {3+5 x}}{6453888 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}+\frac {3474273 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2151296}\\ &=\frac {164 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^5}-\frac {42863 \sqrt {1-2 x} \sqrt {3+5 x}}{41160 (2+3 x)^4}-\frac {29297 \sqrt {1-2 x} \sqrt {3+5 x}}{82320 (2+3 x)^3}-\frac {55277 \sqrt {1-2 x} \sqrt {3+5 x}}{460992 (2+3 x)^2}+\frac {426781 \sqrt {1-2 x} \sqrt {3+5 x}}{6453888 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}-\frac {3474273 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2151296 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 100, normalized size = 0.48 \begin {gather*} \frac {7 \sqrt {5 x+3} \left (-115230870 x^5-180017865 x^4+19738914 x^3+164918884 x^2+95331368 x+16456032\right )-17371365 \sqrt {7-14 x} (3 x+2)^5 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{75295360 \sqrt {1-2 x} (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

(7*Sqrt[3 + 5*x]*(16456032 + 95331368*x + 164918884*x^2 + 19738914*x^3 - 180017865*x^4 - 115230870*x^5) - 1737
1365*Sqrt[7 - 14*x]*(2 + 3*x)^5*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(75295360*Sqrt[1 - 2*x]*(2 + 3*
x)^5)

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IntegrateAlgebraic [A]  time = 0.42, size = 154, normalized size = 0.74 \begin {gather*} -\frac {121 \sqrt {5 x+3} \left (\frac {143565 (1-2 x)^5}{(5 x+3)^5}+\frac {4689790 (1-2 x)^4}{(5 x+3)^4}+\frac {60029312 (1-2 x)^3}{(5 x+3)^3}-\frac {13666590 (1-2 x)^2}{(5 x+3)^2}-\frac {198938285 (1-2 x)}{5 x+3}-24586240\right )}{10756480 \sqrt {1-2 x} \left (\frac {1-2 x}{5 x+3}+7\right )^5}-\frac {3474273 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2151296 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

(-121*Sqrt[3 + 5*x]*(-24586240 + (143565*(1 - 2*x)^5)/(3 + 5*x)^5 + (4689790*(1 - 2*x)^4)/(3 + 5*x)^4 + (60029
312*(1 - 2*x)^3)/(3 + 5*x)^3 - (13666590*(1 - 2*x)^2)/(3 + 5*x)^2 - (198938285*(1 - 2*x))/(3 + 5*x)))/(1075648
0*Sqrt[1 - 2*x]*(7 + (1 - 2*x)/(3 + 5*x))^5) - (3474273*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(215129
6*Sqrt[7])

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fricas [A]  time = 1.10, size = 146, normalized size = 0.70 \begin {gather*} -\frac {17371365 \, \sqrt {7} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\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 (115230870 \, x^{5} + 180017865 \, x^{4} - 19738914 \, x^{3} - 164918884 \, x^{2} - 95331368 \, x - 16456032\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{150590720 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/150590720*(17371365*sqrt(7)*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*arctan(1/14*sq
rt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(115230870*x^5 + 180017865*x^4 - 1973891
4*x^3 - 164918884*x^2 - 95331368*x - 16456032)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(486*x^6 + 1377*x^5 + 1350*x^4 +
360*x^3 - 240*x^2 - 176*x - 32)

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giac [B]  time = 5.76, size = 452, normalized size = 2.16 \begin {gather*} \frac {3474273}{301181440} \, \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 {1936 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{588245 \, {\left (2 \, x - 1\right )}} - \frac {121 \, \sqrt {10} {\left (203039 \, {\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 )}^{9} + 265495440 \, {\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} + 136071290880 \, {\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} - 774949504000 \, {\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 {650054039040000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {2600216156160000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{7529536 \, {\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 )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

3474273/301181440*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1936/588245*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*
x + 5)/(2*x - 1) - 121/7529536*sqrt(10)*(203039*((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)))^9 + 265495440*((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 + 136071290880*((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 - 774949504000*((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 - 650054039040000*(s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2600216156160000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))^2 + 280)^5

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maple [B]  time = 0.02, size = 353, normalized size = 1.69 \begin {gather*} \frac {\left (8442483390 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{5} \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^{5}[phone] \sqrt {7}\, x^{4} \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^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-276344796 \sqrt {-10 x^{2}-x +3}\, x^{3}-4169127600 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2308864376 \sqrt {-10 x^{2}-x +3}\, x^{2}-3057360240 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1334639152 \sqrt {-10 x^{2}-x +3}\, x -555883680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-230384448 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{150590720 \left (3 x +2\right )^{5} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^(5/2)/(-2*x+1)^(3/2)/(3*x+2)^6,x)

[Out]

1/150590720*(8442483390*7^(1/2)*x^6*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^5
*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^4*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^5+6253691400*7^(1/2)*x^3*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^4-4169127600*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))-276344796*(-10*x^2-x+3)^(1/2)*x^3-3057360240*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x
+3)^(1/2))-2308864376*(-10*x^2-x+3)^(1/2)*x^2-555883680*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1
/2))-1334639152*(-10*x^2-x+3)^(1/2)*x-230384448*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(3*x+2)^5/(2
*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [B]  time = 1.25, size = 398, normalized size = 1.90 \begin {gather*} \frac {3474273}{30118144} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2133905 \, x}{9680832 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4998019}{19361664 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{945 \, {\left (243 \, \sqrt {-10 \, x^{2} - x + 3} x^{5} + 810 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 1080 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 720 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 240 \, \sqrt {-10 \, x^{2} - x + 3} x + 32 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {331}{17640 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {83537}{740880 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {23353}{109760 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {137335}{921984 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

3474273/30118144*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2133905/9680832*x/sqrt(-10*x^2 -
x + 3) + 4998019/19361664/sqrt(-10*x^2 - x + 3) + 1/945/(243*sqrt(-10*x^2 - x + 3)*x^5 + 810*sqrt(-10*x^2 - x
+ 3)*x^4 + 1080*sqrt(-10*x^2 - x + 3)*x^3 + 720*sqrt(-10*x^2 - x + 3)*x^2 + 240*sqrt(-10*x^2 - x + 3)*x + 32*s
qrt(-10*x^2 - x + 3)) - 331/17640/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 + 216*sqrt(-10
*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x^2 - x + 3)) + 83537/740880/(27*sqrt(-10*x^2 - x
+ 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) - 23353/10976
0/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 137335/921984/(3*sqrt
(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^6),x)

[Out]

int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^6), 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)**(3/2)/(2+3*x)**6,x)

[Out]

Timed out

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