3.27.1 \(\int \frac {(2+3 x) (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\) [2601]

Optimal. Leaf size=118 \[ -\frac {5975}{64} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5975}{528} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {239 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{7/2}}{33 (1-2 x)^{3/2}}+\frac {13145}{64} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]

[Out]

7/33*(3+5*x)^(7/2)/(1-2*x)^(3/2)+13145/128*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-239/66*(3+5*x)^(5/2)/(
1-2*x)^(1/2)-5975/528*(3+5*x)^(3/2)*(1-2*x)^(1/2)-5975/64*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {79, 49, 52, 56, 222} \begin {gather*} \frac {13145}{64} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {7 (5 x+3)^{7/2}}{33 (1-2 x)^{3/2}}-\frac {239 (5 x+3)^{5/2}}{66 \sqrt {1-2 x}}-\frac {5975}{528} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {5975}{64} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5975*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64 - (5975*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/528 - (239*(3 + 5*x)^(5/2))/(66*
Sqrt[1 - 2*x]) + (7*(3 + 5*x)^(7/2))/(33*(1 - 2*x)^(3/2)) + (13145*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])
/64

Rule 49

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 52

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 56

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 79

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] || L
tQ[p, n]))))

Rule 222

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)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {7 (3+5 x)^{7/2}}{33 (1-2 x)^{3/2}}-\frac {239}{66} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {239 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{7/2}}{33 (1-2 x)^{3/2}}+\frac {5975}{132} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {5975}{528} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {239 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{7/2}}{33 (1-2 x)^{3/2}}+\frac {5975}{32} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {5975}{64} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5975}{528} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {239 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{7/2}}{33 (1-2 x)^{3/2}}+\frac {65725}{128} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {5975}{64} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5975}{528} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {239 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{7/2}}{33 (1-2 x)^{3/2}}+\frac {1}{64} \left (13145 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {5975}{64} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5975}{528} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {239 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{7/2}}{33 (1-2 x)^{3/2}}+\frac {13145}{64} \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 = 0.42, size = 77, normalized size = 0.65 \begin {gather*} \frac {1}{192} \left (-\frac {\sqrt {3+5 x} \left (29601-84064 x+20820 x^2+3600 x^3\right )}{(1-2 x)^{3/2}}-39435 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((Sqrt[3 + 5*x]*(29601 - 84064*x + 20820*x^2 + 3600*x^3))/(1 - 2*x)^(3/2)) - 39435*Sqrt[10]*ArcTan[Sqrt[6 +
10*x]/(Sqrt[11] - Sqrt[5 - 10*x])])/192

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Maple [A]
time = 0.09, size = 137, normalized size = 1.16

method	result	size
default	\(\frac {\left (157740 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-14400 x^{3} \sqrt {-10 x^{2}-x +3}-157740 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -83280 x^{2} \sqrt {-10 x^{2}-x +3}+39435 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+336256 x \sqrt {-10 x^{2}-x +3}-118404 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{768 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768*(157740*10^(1/2)*arcsin(20/11*x+1/11)*x^2-14400*x^3*(-10*x^2-x+3)^(1/2)-157740*10^(1/2)*arcsin(20/11*x+1
/11)*x-83280*x^2*(-10*x^2-x+3)^(1/2)+39435*10^(1/2)*arcsin(20/11*x+1/11)+336256*x*(-10*x^2-x+3)^(1/2)-118404*(
-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (83) = 166\).
time = 0.49, size = 186, normalized size = 1.58 \begin {gather*} \frac {13145}{256} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{4 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{8 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac {385 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{48 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {165 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {4235 \, \sqrt {-10 \, x^{2} - x + 3}}{96 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {43285 \, \sqrt {-10 \, x^{2} - x + 3}}{192 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13145/256*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 7/4*(-10*x^2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x
+ 1) - 3/8*(-10*x^2 - x + 3)^(5/2)/(8*x^3 - 12*x^2 + 6*x - 1) - 385/48*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^
2 + 6*x - 1) + 165/32*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 4235/96*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x +
1) + 43285/192*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]
time = 0.53, size = 102, normalized size = 0.86 \begin {gather*} -\frac {39435 \, \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 (3600 \, x^{3} + 20820 \, x^{2} - 84064 \, x + 29601\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{768 \, {\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)*(3+5*x)^(5/2)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/768*(39435*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*(3600*x^3 + 20820*x^2 - 84064*x + 29601)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*
x + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.57, size = 84, normalized size = 0.71 \begin {gather*} \frac {13145}{128} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 239 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 26290 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 433785 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{4800 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13145/128*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/4800*(4*(3*(12*sqrt(5)*(5*x + 3) + 239*sqrt(5))*(5*
x + 3) - 26290*sqrt(5))*(5*x + 3) + 433785*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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

[Out]

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

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