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

Optimal. Leaf size=140 \[ -\frac {1183 (5 x+3)^{7/2}}{363 \sqrt {1-2 x}}+\frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {24749 \sqrt {1-2 x} (5 x+3)^{5/2}}{2904}-\frac {123745 \sqrt {1-2 x} (5 x+3)^{3/2}}{2112}-\frac {123745}{256} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {272239}{256} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 140, 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 = {89, 78, 50, 54, 216} \begin {gather*} -\frac {1183 (5 x+3)^{7/2}}{363 \sqrt {1-2 x}}+\frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {24749 \sqrt {1-2 x} (5 x+3)^{5/2}}{2904}-\frac {123745 \sqrt {1-2 x} (5 x+3)^{3/2}}{2112}-\frac {123745}{256} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {272239}{256} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-123745*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256 - (123745*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/2112 - (24749*Sqrt[1 - 2*x]
*(3 + 5*x)^(5/2))/2904 + (49*(3 + 5*x)^(7/2))/(66*(1 - 2*x)^(3/2)) - (1183*(3 + 5*x)^(7/2))/(363*Sqrt[1 - 2*x]
) + (272239*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/256

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {(3+5 x)^{5/2} \left (\frac {2069}{2}+297 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=\frac {49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {1183 (3+5 x)^{7/2}}{363 \sqrt {1-2 x}}+\frac {24749}{484} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {24749 \sqrt {1-2 x} (3+5 x)^{5/2}}{2904}+\frac {49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {1183 (3+5 x)^{7/2}}{363 \sqrt {1-2 x}}+\frac {123745}{528} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {123745 \sqrt {1-2 x} (3+5 x)^{3/2}}{2112}-\frac {24749 \sqrt {1-2 x} (3+5 x)^{5/2}}{2904}+\frac {49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {1183 (3+5 x)^{7/2}}{363 \sqrt {1-2 x}}+\frac {123745}{128} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {123745}{256} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {123745 \sqrt {1-2 x} (3+5 x)^{3/2}}{2112}-\frac {24749 \sqrt {1-2 x} (3+5 x)^{5/2}}{2904}+\frac {49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {1183 (3+5 x)^{7/2}}{363 \sqrt {1-2 x}}+\frac {1361195}{512} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {123745}{256} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {123745 \sqrt {1-2 x} (3+5 x)^{3/2}}{2112}-\frac {24749 \sqrt {1-2 x} (3+5 x)^{5/2}}{2904}+\frac {49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {1183 (3+5 x)^{7/2}}{363 \sqrt {1-2 x}}+\frac {1}{256} \left (272239 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {123745}{256} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {123745 \sqrt {1-2 x} (3+5 x)^{3/2}}{2112}-\frac {24749 \sqrt {1-2 x} (3+5 x)^{5/2}}{2904}+\frac {49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {1183 (3+5 x)^{7/2}}{363 \sqrt {1-2 x}}+\frac {272239}{256} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 95, normalized size = 0.68 \begin {gather*} \frac {2 \sqrt {2 x-1} \sqrt {5 x+3} \left (28800 x^4+146160 x^3+497868 x^2-1713440 x+617319\right )+816717 \sqrt {10} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{1536 \sqrt {1-2 x} (2 x-1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(617319 - 1713440*x + 497868*x^2 + 146160*x^3 + 28800*x^4) + 816717*Sqrt[10]*(
1 - 2*x)^2*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(1536*Sqrt[1 - 2*x]*(-1 + 2*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.68, size = 140, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {11-2 (5 x+3)} \left (1152 (5 x+3)^{9/2}+15408 (5 x+3)^{7/2}+296988 (5 x+3)^{5/2}-10889560 (5 x+3)^{3/2}[phone] \sqrt {5 x+3}\right )}{768 \sqrt {5} (2 (5 x+3)-11)^2}-\frac {272239}{128} \sqrt {\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/768*(Sqrt[11 - 2*(3 + 5*x)]*(44919435*Sqrt[3 + 5*x] - 10889560*(3 + 5*x)^(3/2) + 296988*(3 + 5*x)^(5/2) + 1
5408*(3 + 5*x)^(7/2) + 1152*(3 + 5*x)^(9/2)))/(Sqrt[5]*(-11 + 2*(3 + 5*x))^2) - (272239*Sqrt[5/2]*ArcTan[(Sqrt
[2]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/128

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fricas [A]  time = 1.34, size = 107, normalized size = 0.76 \begin {gather*} -\frac {816717 \, \sqrt {5} \sqrt {2} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 4 \, {\left (28800 \, x^{4} + 146160 \, x^{3} + 497868 \, x^{2} - 1713440 \, x + 617319\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3072 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3072*(816717*sqrt(5)*sqrt(2)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2
*x + 1)/(10*x^2 + x - 3)) + 4*(28800*x^4 + 146160*x^3 + 497868*x^2 - 1713440*x + 617319)*sqrt(5*x + 3)*sqrt(-2
*x + 1))/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.19, size = 97, normalized size = 0.69 \begin {gather*} \frac {272239}{512} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 107 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 24749 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 2722390 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 44919435 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{96000 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

272239/512*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/96000*(4*(3*(12*(8*sqrt(5)*(5*x + 3) + 107*sqrt(5)
)*(5*x + 3) + 24749*sqrt(5))*(5*x + 3) - 2722390*sqrt(5))*(5*x + 3) + 44919435*sqrt(5))*sqrt(5*x + 3)*sqrt(-10
*x + 5)/(2*x - 1)^2

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maple [A]  time = 0.01, size = 154, normalized size = 1.10 \begin {gather*} \frac {\left (-115200 \sqrt {-10 x^{2}-x +3}\, x^{4}-584640 \sqrt {-10 x^{2}-x +3}\, x^{3}+3266868 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1991472 \sqrt {-10 x^{2}-x +3}\, x^{2}-3266868 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+6853760 \sqrt {-10 x^{2}-x +3}\, x +816717 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2469276 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{3072 \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((3*x+2)^2*(5*x+3)^(5/2)/(-2*x+1)^(5/2),x)

[Out]

1/3072*(-115200*(-10*x^2-x+3)^(1/2)*x^4+3266868*10^(1/2)*x^2*arcsin(20/11*x+1/11)-584640*(-10*x^2-x+3)^(1/2)*x
^3-3266868*10^(1/2)*x*arcsin(20/11*x+1/11)-1991472*(-10*x^2-x+3)^(1/2)*x^2+816717*10^(1/2)*arcsin(20/11*x+1/11
)+6853760*(-10*x^2-x+3)^(1/2)*x-2469276*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(2*x-1)^2/(-10*x^2-x
+3)^(1/2)

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maxima [B]  time = 1.46, size = 247, normalized size = 1.76 \begin {gather*} \frac {272239}{1024} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {49 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{8 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {21 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{8 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{8 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {5445}{256} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2695 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{96 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {1155 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {165 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{64 \, {\left (2 \, x - 1\right )}} + \frac {29645 \, \sqrt {-10 \, x^{2} - x + 3}}{192 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {104335 \, \sqrt {-10 \, x^{2} - x + 3}}{96 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

272239/1024*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 49/8*(-10*x^2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 -
8*x + 1) - 21/8*(-10*x^2 - x + 3)^(5/2)/(8*x^3 - 12*x^2 + 6*x - 1) - 3/8*(-10*x^2 - x + 3)^(5/2)/(4*x^2 - 4*x
+ 1) - 5445/256*sqrt(-10*x^2 - x + 3) - 2695/96*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 1155/32*(
-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 165/64*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 29645/192*sqrt(-10*x^2 -
x + 3)/(4*x^2 - 4*x + 1) + 104335/96*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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