3.25.70 \(\int \frac {(2+3 x) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\) [2470]

Optimal. Leaf size=94 \[ -\frac {1947}{320} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59}{80} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {21417 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320 \sqrt {10}} \]

[Out]

21417/3200*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-59/80*(3+5*x)^(3/2)*(1-2*x)^(1/2)-1/10*(3+5*x)^(5/2)*(
1-2*x)^(1/2)-1947/320*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222} \begin {gather*} \frac {21417 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320 \sqrt {10}}-\frac {1}{10} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {59}{80} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {1947}{320} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1947*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/320 - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/80 - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/
2))/10 + (21417*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320*Sqrt[10])

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 81

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 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)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {59}{20} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {59}{80} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1947}{160} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {1947}{320} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59}{80} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {21417}{640} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1947}{320} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59}{80} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {21417 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{320 \sqrt {5}}\\ &=-\frac {1947}{320} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59}{80} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {21417 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 73, normalized size = 0.78 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (8829+21135 x+13100 x^2+4000 x^3\right )-21417 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{3200 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*(8829 + 21135*x + 13100*x^2 + 4000*x^3) - 21417*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt
[3 + 5*x]])/(3200*Sqrt[3 + 5*x])

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Maple [A]
time = 0.08, size = 87, normalized size = 0.93

method	result	size
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-16000 x^{2} \sqrt {-10 x^{2}-x +3}+21417 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-42800 x \sqrt {-10 x^{2}-x +3}-58860 \sqrt {-10 x^{2}-x +3}\right )}{6400 \sqrt {-10 x^{2}-x +3}}\)	\(87\)
risch	\(\frac {\left (800 x^{2}+2140 x +2943\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{320 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {21417 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6400 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6400*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-16000*x^2*(-10*x^2-x+3)^(1/2)+21417*10^(1/2)*arcsin(20/11*x+1/11)-42800*x
*(-10*x^2-x+3)^(1/2)-58860*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.64, size = 58, normalized size = 0.62 \begin {gather*} -\frac {5}{2} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {107}{16} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {21417}{6400} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {2943}{320} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2*sqrt(-10*x^2 - x + 3)*x^2 - 107/16*sqrt(-10*x^2 - x + 3)*x - 21417/6400*sqrt(10)*arcsin(-20/11*x - 1/11)
- 2943/320*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 0.39, size = 67, normalized size = 0.71 \begin {gather*} -\frac {1}{320} \, {\left (800 \, x^{2} + 2140 \, x + 2943\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {21417}{6400} \, \sqrt {10} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/320*(800*x^2 + 2140*x + 2943)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 21417/6400*sqrt(10)*arctan(1/20*sqrt(10)*(20*x
+ 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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Sympy [A]
time = 40.51, size = 265, normalized size = 2.82 \begin {gather*} \frac {2 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (\frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{968} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{8}\right )}{8} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{25} + \frac {6 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (\frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {3 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{16} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/968 - sqrt(2)*sqrt(5 - 10*x
)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) <
sqrt(22)/2)))/25 + 6*sqrt(5)*Piecewise((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3*sqrt
(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 5*asin(sqrt(22)*
sqrt(5*x + 3)/11)/16)/16, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/25

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Giac [A]
time = 1.54, size = 54, normalized size = 0.57 \begin {gather*} -\frac {1}{3200} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x + 83\right )} {\left (5 \, x + 3\right )} + 1947\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 21417 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3200*sqrt(5)*(2*(4*(40*x + 83)*(5*x + 3) + 1947)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 21417*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*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 )\,{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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