3.25.1 \(\int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx\) [2401]

Optimal. Leaf size=160 \[ \frac {2767149 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}+\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {30438639 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2560000 \sqrt {10}} \]

[Out]

-63/400*(1-2*x)^(7/2)*(3+5*x)^(3/2)-1/20*(1-2*x)^(7/2)*(3+5*x)^(5/2)[phone]/25600000*arcsin(1/11*22^(1/2)*(3
+5*x)^(1/2))*10^(1/2)+83853/256000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+7623/64000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-2079/640
0*(1-2*x)^(7/2)*(3+5*x)^(1/2)+2767149/2560000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {30438639 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2560000 \sqrt {10}}-\frac {1}{20} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac {63}{400} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {2079 \sqrt {5 x+3} (1-2 x)^{7/2}}{6400}+\frac {7623 \sqrt {5 x+3} (1-2 x)^{5/2}}{64000}+\frac {83853 \sqrt {5 x+3} (1-2 x)^{3/2}}{256000}+\frac {2767149 \sqrt {5 x+3} \sqrt {1-2 x}}{2560000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2767149*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2560000 + (83853*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/256000 + (7623*(1 - 2*x)
^(5/2)*Sqrt[3 + 5*x])/64000 - (2079*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/6400 - (63*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))
/400 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/20 + (30438639*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2560000*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 (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx &=-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {63}{40} \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {2079}{800} \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx\\ &=-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {22869 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{12800}\\ &=\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {83853 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{25600}\\ &=\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {2767149 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{512000}\\ &=\frac {2767149 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}+\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {30438639 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{5120000}\\ &=\frac {2767149 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}+\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {30438639 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2560000 \sqrt {5}}\\ &=\frac {2767149 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}+\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {30438639 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2560000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 88, normalized size = 0.55 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (2152197+34806375 x+36544140 x^2-102392800 x^3-102288000 x^4+119040000 x^5+128000000 x^6\right )-30438639 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{25600000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(2152197 + 34806375*x + 36544140*x^2 - 102392800*x^3 - 102288000*x^4 + 119040000*x^5 + 12800
0000*x^6) - 30438639*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(25600000*Sqrt[3 + 5*x])

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Maple [A]
time = 0.11, size = 138, normalized size = 0.86

method	result	size
risch	\(-\frac {\left (25600000 x^{5}+8448000 x^{4}-25526400 x^{3}-5162720 x^{2}[phone] x +717399\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2560000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {30438639 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{51200000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(113\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (512000000 x^{5} \sqrt {-10 x^{2}-x +3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}-510528000 x^{3} \sqrt {-10 x^{2}-x +3}-103254400 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{51200000 \sqrt {-10 x^{2}-x +3}}\)	\(138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/51200000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(512000000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/2)-51
0528000*x^3*(-10*x^2-x+3)^(1/2)-103254400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)+20812
9200*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.64, size = 99, normalized size = 0.62 \begin {gather*} \frac {1}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {13}{1000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {693}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {693}{32000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {251559}{128000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {30438639}{51200000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {251559}{2560000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(-10*x^2 - x + 3)^(5/2)*x + 13/1000*(-10*x^2 - x + 3)^(5/2) + 693/1600*(-10*x^2 - x + 3)^(3/2)*x + 693/32
000*(-10*x^2 - x + 3)^(3/2) + 251559/128000*sqrt(-10*x^2 - x + 3)*x - 30438639/51200000*sqrt(10)*arcsin(-20/11
*x - 1/11) + 251559/2560000*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 0.50, size = 82, normalized size = 0.51 \begin {gather*} \frac {1}{2560000} \, {\left (25600000 \, x^{5} + 8448000 \, x^{4} - 25526400 \, x^{3} - 5162720 \, x^{2} + 10406460 \, x + 717399\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {30438639}{51200000} \, \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((1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/2560000*(25600000*x^5 + 8448000*x^4 - 25526400*x^3 - 5162720*x^2 + 10406460*x + 717399)*sqrt(5*x + 3)*sqrt(-
2*x + 1) - 30438639/51200000*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 [F(-1)] Timed out
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((1-2*x)**(5/2)*(2+3*x)*(3+5*x)**(3/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (115) = 230\).
time = 1.58, size = 356, normalized size = 2.22 \begin {gather*} \frac {1}{128000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {13}{48000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {137}{9600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {17}{15000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{400} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 15385695)
*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
+ 13/48000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sq
rt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 137/9600000*sqrt(5)*(2*(
4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1
/11*sqrt(22)*sqrt(5*x + 3))) - 17/15000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x +
5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/400*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*
x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt
(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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