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

Optimal. Leaf size=138 \[ \frac {9}{80} \sqrt {1-2 x} (5 x+3)^{7/2}+\frac {49 (5 x+3)^{7/2}}{22 \sqrt {1-2 x}}+\frac {25397 \sqrt {1-2 x} (5 x+3)^{5/2}}{3520}+\frac {25397}{512} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {838101 \sqrt {1-2 x} \sqrt {5 x+3}}{2048}-\frac {9219111 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2048 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 138, 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, 80, 50, 54, 216} \begin {gather*} \frac {9}{80} \sqrt {1-2 x} (5 x+3)^{7/2}+\frac {49 (5 x+3)^{7/2}}{22 \sqrt {1-2 x}}+\frac {25397 \sqrt {1-2 x} (5 x+3)^{5/2}}{3520}+\frac {25397}{512} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {838101 \sqrt {1-2 x} \sqrt {5 x+3}}{2048}-\frac {9219111 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2048 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(838101*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2048 + (25397*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/512 + (25397*Sqrt[1 - 2*x]*(
3 + 5*x)^(5/2))/3520 + (49*(3 + 5*x)^(7/2))/(22*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/80 - (92191
11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2048*Sqrt[10])

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 80

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

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)^{3/2}} \, dx &=\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}-\frac {1}{22} \int \frac {(3+5 x)^{5/2} \left (\frac {1833}{2}+99 x\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {76191 \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx}{1760}\\ &=\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {25397}{128} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {25397}{512} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {838101 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{1024}\\ &=\frac {838101 \sqrt {1-2 x} \sqrt {3+5 x}}{2048}+\frac {25397}{512} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {9219111 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{4096}\\ &=\frac {838101 \sqrt {1-2 x} \sqrt {3+5 x}}{2048}+\frac {25397}{512} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {9219111 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2048 \sqrt {5}}\\ &=\frac {838101 \sqrt {1-2 x} \sqrt {3+5 x}}{2048}+\frac {25397}{512} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {9219111 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2048 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 88, normalized size = 0.64 \begin {gather*} \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (57600 x^4+243520 x^3+517096 x^2+966014 x-1405233\right )-9219111 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{20480 \sqrt {-(1-2 x)^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-1405233 + 966014*x + 517096*x^2 + 243520*x^3 + 57600*x^4) - 9219111*Sqrt[1
0]*(-1 + 2*x)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(20480*Sqrt[-(1 - 2*x)^2])

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IntegrateAlgebraic [A]  time = 0.23, size = 141, normalized size = 1.02 \begin {gather*} \frac {121 \sqrt {5 x+3} \left (\frac {9523875 (1-2 x)^4}{(5 x+3)^4}+\frac {13968350 (1-2 x)^3}{(5 x+3)^3}+\frac {7415924 (1-2 x)^2}{(5 x+3)^2}+\frac {1619720 (1-2 x)}{5 x+3}+100352\right )}{2048 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}+\frac {9219111 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{2048 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(121*Sqrt[3 + 5*x]*(100352 + (9523875*(1 - 2*x)^4)/(3 + 5*x)^4 + (13968350*(1 - 2*x)^3)/(3 + 5*x)^3 + (7415924
*(1 - 2*x)^2)/(3 + 5*x)^2 + (1619720*(1 - 2*x))/(3 + 5*x)))/(2048*Sqrt[1 - 2*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^
4) + (9219111*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(2048*Sqrt[10])

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fricas [A]  time = 1.29, size = 91, normalized size = 0.66 \begin {gather*} \frac {9219111 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (57600 \, x^{4} + 243520 \, x^{3} + 517096 \, x^{2} + 966014 \, x - 1405233\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{40960 \, {\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)^(3/2),x, algorithm="fricas")

[Out]

1/40960*(9219111*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x -
3)) + 20*(57600*x^4 + 243520*x^3 + 517096*x^2 + 966014*x - 1405233)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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giac [A]  time = 0.95, size = 97, normalized size = 0.70 \begin {gather*} -\frac {9219111}{20480} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} + 329 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 25397 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1396835 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 46095555 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{256000 \, {\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)^(3/2),x, algorithm="giac")

[Out]

-9219111/20480*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/256000*(2*(4*(8*(36*sqrt(5)*(5*x + 3) + 329*sq
rt(5))*(5*x + 3) + 25397*sqrt(5))*(5*x + 3) + 1396835*sqrt(5))*(5*x + 3) - 46095555*sqrt(5))*sqrt(5*x + 3)*sqr
t(-10*x + 5)/(2*x - 1)

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maple [A]  time = 0.01, size = 140, normalized size = 1.01 \begin {gather*} -\frac {\left (-1152000 \sqrt {-10 x^{2}-x +3}\, x^{4}-4870400 \sqrt {-10 x^{2}-x +3}\, x^{3}-10341920 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-19320280 \sqrt {-10 x^{2}-x +3}\, x -9219111 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{40960 \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((3*x+2)^2*(5*x+3)^(5/2)/(-2*x+1)^(3/2),x)

[Out]

-1/40960*(-1152000*(-10*x^2-x+3)^(1/2)*x^4-4870400*(-10*x^2-x+3)^(1/2)*x^3+18438222*10^(1/2)*x*arcsin(20/11*x+
1/11)-10341920*(-10*x^2-x+3)^(1/2)*x^2-9219111*10^(1/2)*arcsin(20/11*x+1/11)-19320280*(-10*x^2-x+3)^(1/2)*x+28
104660*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.39, size = 109, normalized size = 0.79 \begin {gather*} -\frac {1125 \, x^{5}}{8 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {21725 \, x^{4}}{32 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {414505 \, x^{3}}{256 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3190679 \, x^{2}}{1024 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {9219111}{40960} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {4128123 \, x}{2048 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4215699}{2048 \, \sqrt {-10 \, x^{2} - x + 3}} \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)^(3/2),x, algorithm="maxima")

[Out]

-1125/8*x^5/sqrt(-10*x^2 - x + 3) - 21725/32*x^4/sqrt(-10*x^2 - x + 3) - 414505/256*x^3/sqrt(-10*x^2 - x + 3)
- 3190679/1024*x^2/sqrt(-10*x^2 - x + 3) + 9219111/40960*sqrt(10)*arcsin(-20/11*x - 1/11) + 4128123/2048*x/sqr
t(-10*x^2 - x + 3) + 4215699/2048/sqrt(-10*x^2 - x + 3)

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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 )}^{3/2}} \,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)^(3/2),x)

[Out]

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

[Out]

Timed out

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