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

Optimal. Leaf size=202 \[ \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac {139745 \sqrt {5 x+3}}{1613472 \sqrt {1-2 x}}-\frac {14135 \sqrt {5 x+3}}{153664 \sqrt {1-2 x} (3 x+2)}-\frac {2013 \sqrt {5 x+3}}{10976 \sqrt {1-2 x} (3 x+2)^2}-\frac {2717 \sqrt {5 x+3}}{8232 \sqrt {1-2 x} (3 x+2)^3}+\frac {43 \sqrt {5 x+3}}{588 \sqrt {1-2 x} (3 x+2)^4}-\frac {547745 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1075648 \sqrt {7}} \]

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Rubi [A]  time = 0.08, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {98, 149, 151, 152, 12, 93, 204} \begin {gather*} \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac {139745 \sqrt {5 x+3}}{1613472 \sqrt {1-2 x}}-\frac {14135 \sqrt {5 x+3}}{153664 \sqrt {1-2 x} (3 x+2)}-\frac {2013 \sqrt {5 x+3}}{10976 \sqrt {1-2 x} (3 x+2)^2}-\frac {2717 \sqrt {5 x+3}}{8232 \sqrt {1-2 x} (3 x+2)^3}+\frac {43 \sqrt {5 x+3}}{588 \sqrt {1-2 x} (3 x+2)^4}-\frac {547745 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1075648 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(139745*Sqrt[3 + 5*x])/(1613472*Sqrt[1 - 2*x]) + (43*Sqrt[3 + 5*x])/(588*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (2717*Sq
rt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2013*Sqrt[3 + 5*x])/(10976*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (1413
5*Sqrt[3 + 5*x])/(153664*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - (5
47745*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1075648*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 152

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 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)^{5/2} (2+3 x)^5} \, dx &=\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac {1}{21} \int \frac {\left (-222-\frac {795 x}{2}\right ) \sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\\ &=\frac {43 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^4}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac {\int \frac {-\frac {59169}{2}-50490 x}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{1764}\\ &=\frac {43 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {2717 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)^3}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac {\int \frac {-\frac {851301}{4}-366795 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{37044}\\ &=\frac {43 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {2717 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)^3}-\frac {2013 \sqrt {3+5 x}}{10976 \sqrt {1-2 x} (2+3 x)^2}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac {\int \frac {-\frac {9255015}{8}-1902285 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{518616}\\ &=\frac {43 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {2717 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)^3}-\frac {2013 \sqrt {3+5 x}}{10976 \sqrt {1-2 x} (2+3 x)^2}-\frac {14135 \sqrt {3+5 x}}{153664 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac {\int \frac {-\frac {70128135}{16}-\frac {13357575 x}{4}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{3630312}\\ &=\frac {139745 \sqrt {3+5 x}}{1613472 \sqrt {1-2 x}}+\frac {43 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {2717 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)^3}-\frac {2013 \sqrt {3+5 x}}{10976 \sqrt {1-2 x} (2+3 x)^2}-\frac {14135 \sqrt {3+5 x}}{153664 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {\int \frac {1138761855}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{139767012}\\ &=\frac {139745 \sqrt {3+5 x}}{1613472 \sqrt {1-2 x}}+\frac {43 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {2717 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)^3}-\frac {2013 \sqrt {3+5 x}}{10976 \sqrt {1-2 x} (2+3 x)^2}-\frac {14135 \sqrt {3+5 x}}{153664 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {547745 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2151296}\\ &=\frac {139745 \sqrt {3+5 x}}{1613472 \sqrt {1-2 x}}+\frac {43 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {2717 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)^3}-\frac {2013 \sqrt {3+5 x}}{10976 \sqrt {1-2 x} (2+3 x)^2}-\frac {14135 \sqrt {3+5 x}}{153664 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {547745 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1075648}\\ &=\frac {139745 \sqrt {3+5 x}}{1613472 \sqrt {1-2 x}}+\frac {43 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {2717 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)^3}-\frac {2013 \sqrt {3+5 x}}{10976 \sqrt {1-2 x} (2+3 x)^2}-\frac {14135 \sqrt {3+5 x}}{153664 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac {547745 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1075648 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 105, normalized size = 0.52 \begin {gather*} -\frac {7 \sqrt {5 x+3} \left (45277380 x^5+82071900 x^4+25673409 x^3-27318504 x^2-18627988 x-2906640\right )-1643235 \sqrt {7-14 x} (2 x-1) (3 x+2)^4 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{22588608 (1-2 x)^{3/2} (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/22588608*(7*Sqrt[3 + 5*x]*(-2906640 - 18627988*x - 27318504*x^2 + 25673409*x^3 + 82071900*x^4 + 45277380*x^
5) - 1643235*Sqrt[7 - 14*x]*(-1 + 2*x)*(2 + 3*x)^4*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/((1 - 2*x)^(
3/2)*(2 + 3*x)^4)

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IntegrateAlgebraic [A]  time = 5.09, size = 251, normalized size = 1.24 \begin {gather*} -\frac {5 \sqrt {11-2 (5 x+3)} \left (9055476 \sqrt {5} (5 x+3)^{11/2}-53760240 \sqrt {5} (5 x+3)^{9/2}-41502915 \sqrt {5} (5 x+3)^{7/2}[phone] \sqrt {5} (5 x+3)^{5/2}[phone] \sqrt {5} (5 x+3)^{3/2}+3615117 \sqrt {5} \sqrt {5 x+3}\right )}{3226944 (2 (5 x+3)-11)^2 (3 (5 x+3)+1)^4}-\frac {547745 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{1075648 \sqrt {7}}-\frac {547745 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{1075648 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[11 - 2*(3 + 5*x)]*(3615117*Sqrt[5]*Sqrt[3 + 5*x] + 38889895*Sqrt[5]*(3 + 5*x)^(3/2) + 148638075*Sqrt[
5]*(3 + 5*x)^(5/2) - 41502915*Sqrt[5]*(3 + 5*x)^(7/2) - 53760240*Sqrt[5]*(3 + 5*x)^(9/2) + 9055476*Sqrt[5]*(3
+ 5*x)^(11/2)))/(3226944*(-11 + 2*(3 + 5*x))^2*(1 + 3*(3 + 5*x))^4) - (547745*ArcTan[(Sqrt[2/(34 + Sqrt[1155])
]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(1075648*Sqrt[7]) - (547745*ArcTan[(Sqrt[68 + 2*Sqrt[11
55]]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(1075648*Sqrt[7])

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fricas [A]  time = 1.77, size = 146, normalized size = 0.72 \begin {gather*} -\frac {1643235 \, \sqrt {7} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, 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 (45277380 \, x^{5} + 82071900 \, x^{4} + 25673409 \, x^{3} - 27318504 \, x^{2} - 18627988 \, x - 2906640\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{45177216 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, 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)^(5/2)/(2+3*x)^5,x, algorithm="fricas")

[Out]

-1/45177216*(1643235*sqrt(7)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*arctan(1/14*sqrt(7)*
(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(45277380*x^5 + 82071900*x^4 + 25673409*x^3 -
27318504*x^2 - 18627988*x - 2906640)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104
*x^2 + 32*x + 16)

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giac [B]  time = 5.40, size = 407, normalized size = 2.01 \begin {gather*} \frac {109549}{30118144} \, \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 {88 \, {\left (100 \, \sqrt {5} {\left (5 \, x + 3\right )} - 627 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1764735 \, {\left (2 \, x - 1\right )}^{2}} - \frac {55 \, \sqrt {10} {\left (79441 \, {\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} + 82486488 \, {\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} + 31196222400 \, {\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 {1487445568000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5949782272000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3764768 \, {\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)^(5/2)/(2+3*x)^5,x, algorithm="giac")

[Out]

109549/30118144*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 88/1764735*(100*sqrt(5)*(5*x + 3) - 627*sqrt(
5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 55/3764768*sqrt(10)*(79441*((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 + 82486488*((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 + 31196222400*((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 + 14874455
68000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 5949782272000*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 = 353, normalized size = 1.75 \begin {gather*} \frac {\left (532408140 \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 )-633883320 \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 )-1149006600 \sqrt {-10 x^{2}-x +3}\, x^{4}-433814040 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-359427726 \sqrt {-10 x^{2}-x +3}\, x^{3}-170896440 \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 ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{45177216 \left (3 x +2\right )^{4} \left (2 x -1\right )^{2} \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)^(5/2)/(3*x+2)^5,x)

[Out]

1/45177216*(532408140*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*arc
tan(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))-633883320*(-10*x^2-x+3)^(1/2)*x^5-433814040*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)
^(1/2))-1149006600*(-10*x^2-x+3)^(1/2)*x^4-170896440*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))-359427726*(-10*x^2-x+3)^(1/2)*x^3+52583520*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+
382459056*(-10*x^2-x+3)^(1/2)*x^2+26291760*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]
2*(-10*x^2-x+3)^(1/2)*x+40692960*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(3*x+2)^4/(2*x-1)^2/(-10*x^
2-x+3)^(1/2)

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maxima [B]  time = 1.35, size = 325, normalized size = 1.61 \begin {gather*} \frac {547745}{15059072} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {698725 \, x}{1613472 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343745}{3226944 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {633875 \, x}{691488 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{2268 \, {\left (81 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 96 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 16 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {331}{31752 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {9313}{98784 \, {\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 {659891}{1778112 \, {\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 {296615}{12446784 \, {\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)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^5,x, algorithm="maxima")

[Out]

547745/15059072*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 698725/1613472*x/sqrt(-10*x^2 - x
+ 3) + 343745/3226944/sqrt(-10*x^2 - x + 3) + 633875/691488*x/(-10*x^2 - x + 3)^(3/2) - 1/2268/(81*(-10*x^2 -
x + 3)^(3/2)*x^4 + 216*(-10*x^2 - x + 3)^(3/2)*x^3 + 216*(-10*x^2 - x + 3)^(3/2)*x^2 + 96*(-10*x^2 - x + 3)^(3
/2)*x + 16*(-10*x^2 - x + 3)^(3/2)) + 331/31752/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x
^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) - 9313/98784/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 1
2*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 659891/1778112/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-1
0*x^2 - x + 3)^(3/2)) + 296615/12446784/(-10*x^2 - x + 3)^(3/2)

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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 )}^{5/2}\,{\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)^(5/2)/((1 - 2*x)^(5/2)*(3*x + 2)^5),x)

[Out]

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

[Out]

Timed out

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