3.2487 \(\int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}+\frac {1948963 \sqrt {1-2 x} \sqrt {5 x+3}}{8297856 (3 x+2)}-\frac {12371 \sqrt {1-2 x} \sqrt {5 x+3}}{592704 (3 x+2)^2}-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{105840 (3 x+2)^3}+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{17640 (3 x+2)^4}-\frac {933031 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{307328 \sqrt {7}} \]

[Out]

-933031/2151296*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1/105*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3
*x)^5+437/17640*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4-14831/105840*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3-12371
/592704*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+1948963/8297856*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.06, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}+\frac {1948963 \sqrt {1-2 x} \sqrt {5 x+3}}{8297856 (3 x+2)}-\frac {12371 \sqrt {1-2 x} \sqrt {5 x+3}}{592704 (3 x+2)^2}-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{105840 (3 x+2)^3}+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{17640 (3 x+2)^4}-\frac {933031 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{307328 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(437*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(17640*(2 + 3*x)^4) - (14831*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105840*(2 + 3*x)^
3) - (12371*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(592704*(2 + 3*x)^2) + (1948963*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8297856
*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(105*(2 + 3*x)^5) - (933031*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/(307328*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}}{\sqrt {1-2 x} (2+3 x)^6} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac {1}{105} \int \frac {\left (-\frac {981}{2}-845 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {437 \sqrt {1-2 x} \sqrt {3+5 x}}{17640 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac {\int \frac {-\frac {263381}{4}-111745 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{8820}\\ &=\frac {437 \sqrt {1-2 x} \sqrt {3+5 x}}{17640 (2+3 x)^4}-\frac {14831 \sqrt {1-2 x} \sqrt {3+5 x}}{105840 (2+3 x)^3}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac {\int \frac {-\frac {2624125}{8}-519085 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{185220}\\ &=\frac {437 \sqrt {1-2 x} \sqrt {3+5 x}}{17640 (2+3 x)^4}-\frac {14831 \sqrt {1-2 x} \sqrt {3+5 x}}{105840 (2+3 x)^3}-\frac {12371 \sqrt {1-2 x} \sqrt {3+5 x}}{592704 (2+3 x)^2}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac {\int \frac {-\frac {28511035}{16}-\frac {2164925 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{2593080}\\ &=\frac {437 \sqrt {1-2 x} \sqrt {3+5 x}}{17640 (2+3 x)^4}-\frac {14831 \sqrt {1-2 x} \sqrt {3+5 x}}{105840 (2+3 x)^3}-\frac {12371 \sqrt {1-2 x} \sqrt {3+5 x}}{592704 (2+3 x)^2}+\frac {1948963 \sqrt {1-2 x} \sqrt {3+5 x}}{8297856 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac {\int -\frac {881714295}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{18151560}\\ &=\frac {437 \sqrt {1-2 x} \sqrt {3+5 x}}{17640 (2+3 x)^4}-\frac {14831 \sqrt {1-2 x} \sqrt {3+5 x}}{105840 (2+3 x)^3}-\frac {12371 \sqrt {1-2 x} \sqrt {3+5 x}}{592704 (2+3 x)^2}+\frac {1948963 \sqrt {1-2 x} \sqrt {3+5 x}}{8297856 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}+\frac {933031 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{614656}\\ &=\frac {437 \sqrt {1-2 x} \sqrt {3+5 x}}{17640 (2+3 x)^4}-\frac {14831 \sqrt {1-2 x} \sqrt {3+5 x}}{105840 (2+3 x)^3}-\frac {12371 \sqrt {1-2 x} \sqrt {3+5 x}}{592704 (2+3 x)^2}+\frac {1948963 \sqrt {1-2 x} \sqrt {3+5 x}}{8297856 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}+\frac {933031 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{307328}\\ &=\frac {437 \sqrt {1-2 x} \sqrt {3+5 x}}{17640 (2+3 x)^4}-\frac {14831 \sqrt {1-2 x} \sqrt {3+5 x}}{105840 (2+3 x)^3}-\frac {12371 \sqrt {1-2 x} \sqrt {3+5 x}}{592704 (2+3 x)^2}+\frac {1948963 \sqrt {1-2 x} \sqrt {3+5 x}}{8297856 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac {933031 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{307328 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 84, normalized size = 0.47 \[ \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (87703335 x^4+231277650 x^3+222865988 x^2+93291272 x+14330592\right )}{(3 x+2)^5}-13995465 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{32269440} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(14330592 + 93291272*x + 222865988*x^2 + 231277650*x^3 + 87703335*x^4))/(2 + 3
*x)^5 - 13995465*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/32269440

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fricas [A]  time = 1.07, size = 131, normalized size = 0.73 \[ -\frac {13995465 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (87703335 \, x^{4} + 231277650 \, x^{3} + 222865988 \, x^{2} + 93291272 \, x + 14330592\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{64538880 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64538880*(13995465*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x
+ 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(87703335*x^4 + 231277650*x^3 + 222865988*x^2 + 9329
1272*x + 14330592)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [B]  time = 3.26, size = 426, normalized size = 2.37 \[ \frac {933031}{43025920} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (2103 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 2747920 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 1406935040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 74141312000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {10228753920000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {40915015680000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{460992 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

933031/43025920*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)))) - 1331/460992*sqrt(10)*(2103*((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 + 2747920*((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 + 140693
5040*((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 - 74141312000*((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 - 10228753920000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 40915015680000*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)^5

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maple [B]  time = 0.02, size = 298, normalized size = 1.66 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (3400897995 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{64538880 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64538880*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(3400897995*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]*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+10076734800*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+3358911600*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+447854880*7^(1/2)*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+200628288*(-10*x^2-x+3)^(1/2))/(-10*x^2-x
+3)^(1/2)/(3*x+2)^5

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maxima [A]  time = 1.35, size = 184, normalized size = 1.02 \[ \frac {933031}{4302592} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{315 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {239 \, \sqrt {-10 \, x^{2} - x + 3}}{5880 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {14831 \, \sqrt {-10 \, x^{2} - x + 3}}{105840 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {12371 \, \sqrt {-10 \, x^{2} - x + 3}}{592704 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1948963 \, \sqrt {-10 \, x^{2} - x + 3}}{8297856 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

933031/4302592*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/315*sqrt(-10*x^2 - x + 3)/(243*x^
5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 239/5880*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 +
96*x + 16) - 14831/105840*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) - 12371/592704*sqrt(-10*x^2 - x +
3)/(9*x^2 + 12*x + 4) + 1948963/8297856*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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