3.2592 \(\int \frac {(2+3 x) (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ \frac {7 (5 x+3)^{5/2}}{33 (1-2 x)^{3/2}}-\frac {169 (5 x+3)^{3/2}}{66 \sqrt {1-2 x}}-\frac {845}{88} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {169}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

[Out]

7/33*(3+5*x)^(5/2)/(1-2*x)^(3/2)+169/16*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-169/66*(3+5*x)^(3/2)/(1-2
*x)^(1/2)-845/88*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {78, 47, 50, 54, 216} \[ \frac {7 (5 x+3)^{5/2}}{33 (1-2 x)^{3/2}}-\frac {169 (5 x+3)^{3/2}}{66 \sqrt {1-2 x}}-\frac {845}{88} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {169}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-845*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/88 - (169*(3 + 5*x)^(3/2))/(66*Sqrt[1 - 2*x]) + (7*(3 + 5*x)^(5/2))/(33*(1
- 2*x)^(3/2)) + (169*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/8

Rule 47

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

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 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) (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}-\frac {169}{66} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {169 (3+5 x)^{3/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}+\frac {845}{44} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {845}{88} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {169 (3+5 x)^{3/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}+\frac {845}{16} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {845}{88} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {169 (3+5 x)^{3/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}+\frac {1}{8} \left (169 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {845}{88} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {169 (3+5 x)^{3/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}+\frac {169}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 56, normalized size = 0.58 \[ \frac {1859 \sqrt {22} (2 x-1) \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+56 (5 x+3)^{5/2}}{264 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(56*(3 + 5*x)^(5/2) + 1859*Sqrt[22]*(-1 + 2*x)*Hypergeometric2F1[-3/2, -1/2, 1/2, (-5*(-1 + 2*x))/11])/(264*(1
- 2*x)^(3/2))

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fricas [A]  time = 0.94, size = 97, normalized size = 1.01 \[ -\frac {507 \, \sqrt {5} \sqrt {2} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 4 \, {\left (180 \, x^{2} - 1136 \, x + 369\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{96 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(507*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*(180*x^2 - 1136*x + 369)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.14, size = 71, normalized size = 0.74 \[ \frac {169}{16} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9 \, \sqrt {5} {\left (5 \, x + 3\right )} - 338 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 5577 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{600 \, {\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

169/16*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/600*(4*(9*sqrt(5)*(5*x + 3) - 338*sqrt(5))*(5*x + 3) +
5577*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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maple [A]  time = 0.01, size = 120, normalized size = 1.25 \[ \frac {\left (2028 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-720 \sqrt {-10 x^{2}-x +3}\, x^{2}-2028 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+4544 \sqrt {-10 x^{2}-x +3}\, x +507 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1476 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{96 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(2028*10^(1/2)*x^2*arcsin(20/11*x+1/11)-2028*10^(1/2)*x*arcsin(20/11*x+1/11)-720*(-10*x^2-x+3)^(1/2)*x^2+
507*10^(1/2)*arcsin(20/11*x+1/11)+4544*(-10*x^2-x+3)^(1/2)*x-1476*(-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 [A]  time = 1.18, size = 119, normalized size = 1.24 \[ \frac {169}{32} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{12 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {77 \, \sqrt {-10 \, x^{2} - x + 3}}{24 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {271 \, \sqrt {-10 \, x^{2} - x + 3}}{12 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

169/32*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 7/12*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 3/4*
(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 77/24*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 271/12*sqrt(-10*x^
2 - x + 3)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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