3.6.51 \(\int x^{5/2} (a-b x)^{5/2} \, dx\) [551]

Optimal. Leaf size=171 \[ -\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {5 a^6 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}} \]

[Out]

1/12*a*x^(7/2)*(-b*x+a)^(3/2)+1/6*x^(7/2)*(-b*x+a)^(5/2)+5/512*a^6*arctan(b^(1/2)*x^(1/2)/(-b*x+a)^(1/2))/b^(7
/2)-5/768*a^4*x^(3/2)*(-b*x+a)^(1/2)/b^2-1/192*a^3*x^(5/2)*(-b*x+a)^(1/2)/b+1/32*a^2*x^(7/2)*(-b*x+a)^(1/2)-5/
512*a^5*x^(1/2)*(-b*x+a)^(1/2)/b^3

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Rubi [A]
time = 0.04, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \begin {gather*} \frac {5 a^6 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}}-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a - b*x)^(5/2),x]

[Out]

(-5*a^5*Sqrt[x]*Sqrt[a - b*x])/(512*b^3) - (5*a^4*x^(3/2)*Sqrt[a - b*x])/(768*b^2) - (a^3*x^(5/2)*Sqrt[a - b*x
])/(192*b) + (a^2*x^(7/2)*Sqrt[a - b*x])/32 + (a*x^(7/2)*(a - b*x)^(3/2))/12 + (x^(7/2)*(a - b*x)^(5/2))/6 + (
5*a^6*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(512*b^(7/2))

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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int x^{5/2} (a-b x)^{5/2} \, dx &=\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {1}{12} (5 a) \int x^{5/2} (a-b x)^{3/2} \, dx\\ &=\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {1}{8} a^2 \int x^{5/2} \sqrt {a-b x} \, dx\\ &=\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {1}{64} a^3 \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx\\ &=-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^4\right ) \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx}{384 b}\\ &=-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^5\right ) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{512 b^2}\\ &=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^6\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{1024 b^3}\\ &=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^6\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{512 b^3}\\ &=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^6\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^3}\\ &=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {5 a^6 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 114, normalized size = 0.67 \begin {gather*} \frac {\frac {\sqrt {x} \sqrt {a-b x} \left (-15 a^5-10 a^4 b x-8 a^3 b^2 x^2+432 a^2 b^3 x^3-640 a b^4 x^4+256 b^5 x^5\right )}{b^3}+\frac {15 a^6 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{(-b)^{7/2}}}{1536} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a - b*x)^(5/2),x]

[Out]

((Sqrt[x]*Sqrt[a - b*x]*(-15*a^5 - 10*a^4*b*x - 8*a^3*b^2*x^2 + 432*a^2*b^3*x^3 - 640*a*b^4*x^4 + 256*b^5*x^5)
)/b^3 + (15*a^6*Log[-(Sqrt[-b]*Sqrt[x]) + Sqrt[a - b*x]])/(-b)^(7/2))/1536

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 173.37, size = 349, normalized size = 2.04 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-15 a^{\frac {17}{2}} b^6 \text {ArcCosh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {-a+b x}{a}\right )^{\frac {5}{2}}+15 a^6 b^{\frac {13}{2}} \sqrt {x} \left (-a+b x\right )^2-5 a^5 b^{\frac {15}{2}} x^{\frac {3}{2}} \left (-a+b x\right )^2-2 a^4 b^{\frac {17}{2}} x^{\frac {5}{2}} \left (-a+b x\right )^2+8 a b^{\frac {19}{2}} x^{\frac {7}{2}} \left (-55 a^2+134 a b x-112 b^2 x^2\right ) \left (-a+b x\right )^2+256 b^{\frac {25}{2}} x^{\frac {13}{2}} \left (-a+b x\right )^2\right )}{1536 a^{\frac {5}{2}} b^{\frac {19}{2}} \left (\frac {-a+b x}{a}\right )^{\frac {5}{2}}},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {5 a^6 \text {ArcSin}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{512 b^{\frac {7}{2}}}-\frac {5 a^{\frac {11}{2}} \sqrt {x}}{512 b^3 \sqrt {1-\frac {b x}{a}}}+\frac {5 a^{\frac {9}{2}} x^{\frac {3}{2}}}{1536 b^2 \sqrt {1-\frac {b x}{a}}}+\frac {a^{\frac {7}{2}} x^{\frac {5}{2}}}{768 b \sqrt {1-\frac {b x}{a}}}+\frac {55 a^{\frac {5}{2}} x^{\frac {7}{2}}}{192 \sqrt {1-\frac {b x}{a}}}-\frac {67 a^{\frac {3}{2}} b x^{\frac {9}{2}}}{96 \sqrt {1-\frac {b x}{a}}}+\frac {7 \sqrt {a} b^2 x^{\frac {11}{2}}}{12 \sqrt {1-\frac {b x}{a}}}-\frac {b^3 x^{\frac {13}{2}}}{6 \sqrt {a} \sqrt {1-\frac {b x}{a}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^(5/2)*(a - b*x)^(5/2),x]')

[Out]

Piecewise[{{I / 1536 (-15 a ^ (17 / 2) b ^ 6 ArcCosh[Sqrt[b] Sqrt[x] / Sqrt[a]] ((-a + b x) / a) ^ (5 / 2) + 1
5 a ^ 6 b ^ (13 / 2) Sqrt[x] (-a + b x) ^ 2 - 5 a ^ 5 b ^ (15 / 2) x ^ (3 / 2) (-a + b x) ^ 2 - 2 a ^ 4 b ^ (1
7 / 2) x ^ (5 / 2) (-a + b x) ^ 2 + 8 a b ^ (19 / 2) x ^ (7 / 2) (-55 a ^ 2 + 134 a b x - 112 b ^ 2 x ^ 2) (-a
+ b x) ^ 2 + 256 b ^ (25 / 2) x ^ (13 / 2) (-a + b x) ^ 2) / (a ^ (5 / 2) b ^ (19 / 2) ((-a + b x) / a) ^ (5
/ 2)), Abs[b x / a] > 1}}, 5 a ^ 6 ArcSin[Sqrt[b] Sqrt[x] / Sqrt[a]] / (512 b ^ (7 / 2)) - 5 a ^ (11 / 2) Sqrt
[x] / (512 b ^ 3 Sqrt[1 - b x / a]) + 5 a ^ (9 / 2) x ^ (3 / 2) / (1536 b ^ 2 Sqrt[1 - b x / a]) + a ^ (7 / 2)
x ^ (5 / 2) / (768 b Sqrt[1 - b x / a]) + 55 a ^ (5 / 2) x ^ (7 / 2) / (192 Sqrt[1 - b x / a]) - 67 a ^ (3 /
2) b x ^ (9 / 2) / (96 Sqrt[1 - b x / a]) + 7 Sqrt[a] b ^ 2 x ^ (11 / 2) / (12 Sqrt[1 - b x / a]) - b ^ 3 x ^
(13 / 2) / (6 Sqrt[a] Sqrt[1 - b x / a])]

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Maple [A]
time = 0.11, size = 169, normalized size = 0.99

method	result	size
risch	\(-\frac {\left (-256 b^{5} x^{5}+640 a \,b^{4} x^{4}-432 a^{2} b^{3} x^{3}+8 a^{3} b^{2} x^{2}+10 a^{4} b x +15 a^{5}\right ) \sqrt {x}\, \sqrt {-b x +a}}{1536 b^{3}}+\frac {5 a^{6} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{1024 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {-b x +a}}\)	\(124\)
default	\(-\frac {x^{\frac {5}{2}} \left (-b x +a \right )^{\frac {7}{2}}}{6 b}+\frac {5 a \left (-\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {7}{2}}}{5 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {7}{2}}}{4 b}+\frac {a \left (\frac {\left (-b x +a \right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 a \left (\frac {\left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )}{12 b}\)	\(169\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(-b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/6/b*x^(5/2)*(-b*x+a)^(7/2)+5/12*a/b*(-1/5/b*x^(3/2)*(-b*x+a)^(7/2)+3/10*a/b*(-1/4/b*x^(1/2)*(-b*x+a)^(7/2)+
1/8*a/b*(1/3*(-b*x+a)^(5/2)*x^(1/2)+5/6*a*(1/2*(-b*x+a)^(3/2)*x^(1/2)+3/4*a*(x^(1/2)*(-b*x+a)^(1/2)+1/2*a*(x*(
-b*x+a))^(1/2)/(-b*x+a)^(1/2)/x^(1/2)/b^(1/2)*arctan(b^(1/2)*(x-1/2*a/b)/(-b*x^2+a*x)^(1/2)))))))

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Maxima [A]
time = 0.37, size = 242, normalized size = 1.42 \begin {gather*} -\frac {5 \, a^{6} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{512 \, b^{\frac {7}{2}}} + \frac {\frac {15 \, \sqrt {-b x + a} a^{6} b^{5}}{\sqrt {x}} + \frac {85 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{6} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{6} b^{3}}{x^{\frac {5}{2}}} - \frac {198 \, {\left (-b x + a\right )}^{\frac {7}{2}} a^{6} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (-b x + a\right )}^{\frac {9}{2}} a^{6} b}{x^{\frac {9}{2}}} - \frac {15 \, {\left (-b x + a\right )}^{\frac {11}{2}} a^{6}}{x^{\frac {11}{2}}}}{1536 \, {\left (b^{9} - \frac {6 \, {\left (b x - a\right )} b^{8}}{x} + \frac {15 \, {\left (b x - a\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x - a\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x - a\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x - a\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x - a\right )}^{6} b^{3}}{x^{6}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/512*a^6*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/b^(7/2) + 1/1536*(15*sqrt(-b*x + a)*a^6*b^5/sqrt(x) + 85*(
-b*x + a)^(3/2)*a^6*b^4/x^(3/2) + 198*(-b*x + a)^(5/2)*a^6*b^3/x^(5/2) - 198*(-b*x + a)^(7/2)*a^6*b^2/x^(7/2)
- 85*(-b*x + a)^(9/2)*a^6*b/x^(9/2) - 15*(-b*x + a)^(11/2)*a^6/x^(11/2))/(b^9 - 6*(b*x - a)*b^8/x + 15*(b*x -
a)^2*b^7/x^2 - 20*(b*x - a)^3*b^6/x^3 + 15*(b*x - a)^4*b^5/x^4 - 6*(b*x - a)^5*b^4/x^5 + (b*x - a)^6*b^3/x^6)

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Fricas [A]
time = 0.32, size = 208, normalized size = 1.22 \begin {gather*} \left [-\frac {15 \, a^{6} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (256 \, b^{6} x^{5} - 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} - 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x - 15 \, a^{5} b\right )} \sqrt {-b x + a} \sqrt {x}}{3072 \, b^{4}}, -\frac {15 \, a^{6} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (256 \, b^{6} x^{5} - 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} - 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x - 15 \, a^{5} b\right )} \sqrt {-b x + a} \sqrt {x}}{1536 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/3072*(15*a^6*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) - 2*(256*b^6*x^5 - 640*a*b^5*x^4
+ 432*a^2*b^4*x^3 - 8*a^3*b^3*x^2 - 10*a^4*b^2*x - 15*a^5*b)*sqrt(-b*x + a)*sqrt(x))/b^4, -1/1536*(15*a^6*sqr
t(b)*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) - (256*b^6*x^5 - 640*a*b^5*x^4 + 432*a^2*b^4*x^3 - 8*a^3*b^3*x^2
- 10*a^4*b^2*x - 15*a^5*b)*sqrt(-b*x + a)*sqrt(x))/b^4]

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Sympy [A]
time = 177.75, size = 435, normalized size = 2.54 \begin {gather*} \begin {cases} \frac {5 i a^{\frac {11}{2}} \sqrt {x}}{512 b^{3} \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{\frac {9}{2}} x^{\frac {3}{2}}}{1536 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{\frac {7}{2}} x^{\frac {5}{2}}}{768 b \sqrt {-1 + \frac {b x}{a}}} - \frac {55 i a^{\frac {5}{2}} x^{\frac {7}{2}}}{192 \sqrt {-1 + \frac {b x}{a}}} + \frac {67 i a^{\frac {3}{2}} b x^{\frac {9}{2}}}{96 \sqrt {-1 + \frac {b x}{a}}} - \frac {7 i \sqrt {a} b^{2} x^{\frac {11}{2}}}{12 \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{6} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{512 b^{\frac {7}{2}}} + \frac {i b^{3} x^{\frac {13}{2}}}{6 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {5 a^{\frac {11}{2}} \sqrt {x}}{512 b^{3} \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{\frac {9}{2}} x^{\frac {3}{2}}}{1536 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {a^{\frac {7}{2}} x^{\frac {5}{2}}}{768 b \sqrt {1 - \frac {b x}{a}}} + \frac {55 a^{\frac {5}{2}} x^{\frac {7}{2}}}{192 \sqrt {1 - \frac {b x}{a}}} - \frac {67 a^{\frac {3}{2}} b x^{\frac {9}{2}}}{96 \sqrt {1 - \frac {b x}{a}}} + \frac {7 \sqrt {a} b^{2} x^{\frac {11}{2}}}{12 \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{6} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{512 b^{\frac {7}{2}}} - \frac {b^{3} x^{\frac {13}{2}}}{6 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(-b*x+a)**(5/2),x)

[Out]

Piecewise((5*I*a**(11/2)*sqrt(x)/(512*b**3*sqrt(-1 + b*x/a)) - 5*I*a**(9/2)*x**(3/2)/(1536*b**2*sqrt(-1 + b*x/
a)) - I*a**(7/2)*x**(5/2)/(768*b*sqrt(-1 + b*x/a)) - 55*I*a**(5/2)*x**(7/2)/(192*sqrt(-1 + b*x/a)) + 67*I*a**(
3/2)*b*x**(9/2)/(96*sqrt(-1 + b*x/a)) - 7*I*sqrt(a)*b**2*x**(11/2)/(12*sqrt(-1 + b*x/a)) - 5*I*a**6*acosh(sqrt
(b)*sqrt(x)/sqrt(a))/(512*b**(7/2)) + I*b**3*x**(13/2)/(6*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (-5*a**(
11/2)*sqrt(x)/(512*b**3*sqrt(1 - b*x/a)) + 5*a**(9/2)*x**(3/2)/(1536*b**2*sqrt(1 - b*x/a)) + a**(7/2)*x**(5/2)
/(768*b*sqrt(1 - b*x/a)) + 55*a**(5/2)*x**(7/2)/(192*sqrt(1 - b*x/a)) - 67*a**(3/2)*b*x**(9/2)/(96*sqrt(1 - b*
x/a)) + 7*sqrt(a)*b**2*x**(11/2)/(12*sqrt(1 - b*x/a)) + 5*a**6*asin(sqrt(b)*sqrt(x)/sqrt(a))/(512*b**(7/2)) -
b**3*x**(13/2)/(6*sqrt(a)*sqrt(1 - b*x/a)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (125) = 250\).
time = 0.02, size = 568, normalized size = 3.32 \begin {gather*} 2 b^{2} \left (2 \left (\left (\left (\left (\left (\frac {\frac {1}{174182400}\cdot 7257600 b^{10} \sqrt {x} \sqrt {x}}{b^{10}}-\frac {\frac {1}{174182400}\cdot 725760 b^{9} a}{b^{10}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{174182400}\cdot 816480 b^{8} a^{2}}{b^{10}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{174182400}\cdot 952560 b^{7} a^{3}}{b^{10}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{174182400}\cdot 1190700 b^{6} a^{4}}{b^{10}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{174182400}\cdot 1786050 b^{5} a^{5}}{b^{10}}\right ) \sqrt {x} \sqrt {a-b x}-\frac {42 a^{6} \ln \left |\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right |}{2048 b^{5} \sqrt {-b}}\right )-4 a b \left (2 \left (\left (\left (\left (\frac {\frac {1}{1612800}\cdot 80640 b^{8} \sqrt {x} \sqrt {x}}{b^{8}}-\frac {\frac {1}{1612800}\cdot 10080 b^{7} a}{b^{8}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{1612800}\cdot 11760 b^{6} a^{2}}{b^{8}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{1612800}\cdot 14700 b^{5} a^{3}}{b^{8}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{1612800}\cdot 22050 b^{4} a^{4}}{b^{8}}\right ) \sqrt {x} \sqrt {a-b x}-\frac {14 a^{5} \ln \left |\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right |}{512 b^{4} \sqrt {-b}}\right )+2 a^{2} \left (2 \left (\left (\left (\frac {\frac {1}{23040}\cdot 1440 b^{6} \sqrt {x} \sqrt {x}}{b^{6}}-\frac {\frac {1}{23040}\cdot 240 b^{5} a}{b^{6}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{23040}\cdot 300 b^{4} a^{2}}{b^{6}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{23040}\cdot 450 b^{3} a^{3}}{b^{6}}\right ) \sqrt {x} \sqrt {a-b x}-\frac {10 a^{4} \ln \left |\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right |}{256 b^{3} \sqrt {-b}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-b*x+a)^(5/2),x)

[Out]

1/192*((2*(4*(6*x - a/b)*x - 5*a^2/b^2)*x - 15*a^3/b^3)*sqrt(-b*x + a)*sqrt(x) - 15*a^4*log(abs(-sqrt(-b)*sqrt
(x) + sqrt(-b*x + a)))/(sqrt(-b)*b^3))*a^2 - 1/960*((2*(4*(6*(8*x - a/b)*x - 7*a^2/b^2)*x - 35*a^3/b^3)*x - 10
5*a^4/b^4)*sqrt(-b*x + a)*sqrt(x) - 105*a^5*log(abs(-sqrt(-b)*sqrt(x) + sqrt(-b*x + a)))/(sqrt(-b)*b^4))*a*b +
1/7680*((2*(4*(2*(8*(10*x - a/b)*x - 9*a^2/b^2)*x - 21*a^3/b^3)*x - 105*a^4/b^4)*x - 315*a^5/b^5)*sqrt(-b*x +
a)*sqrt(x) - 315*a^6*log(abs(-sqrt(-b)*sqrt(x) + sqrt(-b*x + a)))/(sqrt(-b)*b^5))*b^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(a - b*x)^(5/2),x)

[Out]

int(x^(5/2)*(a - b*x)^(5/2), x)

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