3.25.79 \(\int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\) [2479]

Optimal. Leaf size=143 \[ -\frac {9458207 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {104040277 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400 \sqrt {10}} \]

[Out]

104040277/1024000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-859837/76800*(3+5*x)^(3/2)*(1-2*x)^(1/2)-78167/
48000*(3+5*x)^(5/2)*(1-2*x)^(1/2)-963/4000*(3+5*x)^(7/2)*(1-2*x)^(1/2)-3/50*(2+3*x)*(3+5*x)^(7/2)*(1-2*x)^(1/2
)-9458207/102400*(1-2*x)^(1/2)*(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 143, 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 = {92, 81, 52, 56, 222} \begin {gather*} \frac {104040277 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{102400 \sqrt {10}}-\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}-\frac {963 \sqrt {1-2 x} (5 x+3)^{7/2}}{4000}-\frac {78167 \sqrt {1-2 x} (5 x+3)^{5/2}}{48000}-\frac {859837 \sqrt {1-2 x} (5 x+3)^{3/2}}{76800}-\frac {9458207 \sqrt {1-2 x} \sqrt {5 x+3}}{102400} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9458207*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400 - (859837*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/76800 - (78167*Sqrt[1 -
2*x]*(3 + 5*x)^(5/2))/48000 - (963*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/4000 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)
^(7/2))/50 + (104040277*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400*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 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}-\frac {1}{50} \int \frac {\left (-314-\frac {963 x}{2}\right ) (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {78167 \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx}{8000}\\ &=-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {859837 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{19200}\\ &=-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {9458207 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{51200}\\ &=-\frac {9458207 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {104040277 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{204800}\\ &=-\frac {9458207 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {104040277 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{102400 \sqrt {5}}\\ &=-\frac {9458207 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {104040277 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 83, normalized size = 0.58 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (138561867+376912905 x+378014260 x^2+303416800 x^3+152208000 x^4+34560000 x^5\right )-312120831 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{3072000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*(138561867 + 376912905*x + 378014260*x^2 + 303416800*x^3 + 152208000*x^4 + 34560000*x^5) -
312120831*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(3072000*Sqrt[3 + 5*x])

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 121, normalized size = 0.85

method	result	size
risch	\(\frac {\left (6912000 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307200 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {104040277 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2048000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(108\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-138240000 x^{4} \sqrt {-10 x^{2}-x +3}-525888000 x^{3} \sqrt {-10 x^{2}-x +3}-898134400 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-973176400 x \sqrt {-10 x^{2}-x +3}-923745780 \sqrt {-10 x^{2}-x +3}\right )}{6144000 \sqrt {-10 x^{2}-x +3}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6144000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-138240000*x^4*(-10*x^2-x+3)^(1/2)-525888000*x^3*(-10*x^2-x+3)^(1/2)-89
8134400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-973176400*x*(-10*x^2-x+3)^(1/2)-923745
780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.53, size = 92, normalized size = 0.64 \begin {gather*} -\frac {45}{2} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {2739}{32} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {280667}{1920} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {2432941}{15360} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {104040277}{2048000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {15395763}{102400} \, \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)^(1/2),x, algorithm="maxima")

[Out]

-45/2*sqrt(-10*x^2 - x + 3)*x^4 - 2739/32*sqrt(-10*x^2 - x + 3)*x^3 - 280667/1920*sqrt(-10*x^2 - x + 3)*x^2 -
2432941/15360*sqrt(-10*x^2 - x + 3)*x - 104040277/2048000*sqrt(10)*arcsin(-20/11*x - 1/11) - 15395763/102400*s
qrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 77, normalized size = 0.54 \begin {gather*} -\frac {1}{307200} \, {\left (6912000 \, x^{4} + 26294400 \, x^{3} + 44906720 \, x^{2} + 48658820 \, x + 46187289\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {104040277}{2048000} \, \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)^2*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/307200*(6912000*x^4 + 26294400*x^3 + 44906720*x^2 + 48658820*x + 46187289)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1
04040277/2048000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

Sympy [A]
time = 170.45, size = 558, normalized size = 3.90 \begin {gather*} \frac {2 \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 )}{125} + \frac {12 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \cdot \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} + \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{32} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{125} + \frac {18 \sqrt {5} \left (\begin {cases} \frac {161051 \sqrt {2} \left (- \frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} + \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{1331} + \frac {15 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{7744} + \frac {5 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{3748096} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {63 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{256}\right )}{64} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{125} \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)**(1/2),x)

[Out]

2*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)/1
6)/16, (sqrt(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/125 + 12*sqrt(5)*Piecewise((14641*sqrt(2
)*(2*sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 7*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872
+ sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 - sqr
t(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 35*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)/32, (sqrt(5*x + 3) > -sqrt(22)/
2) & (sqrt(5*x + 3) < sqrt(22)/2)))/125 + 18*sqrt(5)*Piecewise((161051*sqrt(2)*(-2*sqrt(2)*(5 - 10*x)**(5/2)*(
5*x + 3)**(5/2)/805255 + sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/1331 + 15*sqrt(2)*sqrt(5 - 10*x)*(-20*x -
1)*sqrt(5*x + 3)/7744 + 5*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**
2 - 5929)/3748096 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 63*asin(sqrt(22)*sqrt(5*x + 3)/11)/256)/64, (sqr
t(5*x + 3) > -sqrt(22)/2) & (sqrt(5*x + 3) < sqrt(22)/2)))/125

________________________________________________________________________________________

Giac [A]
time = 2.65, size = 72, normalized size = 0.50 \begin {gather*} -\frac {1}{15360000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (36 \, {\left (240 \, x + 481\right )} {\left (5 \, x + 3\right )} + 78167\right )} {\left (5 \, x + 3\right )} + 4299185\right )} {\left (5 \, x + 3\right )} + 141873105\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 1560604155 \, \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)^2*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1/15360000*sqrt(5)*(2*(4*(8*(36*(240*x + 481)*(5*x + 3) + 78167)*(5*x + 3) + 4299185)*(5*x + 3) + 141873105)*
sqrt(5*x + 3)*sqrt(-10*x + 5) - 1560604155*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))

________________________________________________________________________________________

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}}{\sqrt {1-2\,x}} \,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)^(1/2),x)

[Out]

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

________________________________________________________________________________________