3.6.45 \(\int x^{5/2} (a+b x)^{5/2} \, dx\) [545]

Optimal. Leaf size=164 \[ \frac {5 a^5 \sqrt {x} \sqrt {a+b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {a^3 x^{5/2} \sqrt {a+b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a+b x}+\frac {1}{12} a x^{7/2} (a+b x)^{3/2}+\frac {1}{6} x^{7/2} (a+b x)^{5/2}-\frac {5 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}} \]

[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*arctanh(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 = 164, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {52, 65, 223, 212} \begin {gather*} -\frac {5 a^6 \tanh ^{-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*ArcTanh[(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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 \tanh ^{-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 = 0.11, size = 110, normalized size = 0.67 \begin {gather*} \frac {\sqrt {b} \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 )+15 a^6 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{1536 b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x)^(5/2),x]

[Out]

(Sqrt[b]*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) + 15*a^6*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/(1536*b^(7/2))

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Mathics [A]
time = 174.24, size = 154, normalized size = 0.94 \begin {gather*} \frac {-15 a^{\frac {17}{2}} b^6 \text {ArcSinh}\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}{1536 a^{\frac {5}{2}} b^{\frac {19}{2}} \left (\frac {a+b x}{a}\right )^{\frac {5}{2}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-15 a ^ (17 / 2) b ^ 6 ArcSinh[Sqrt[b] Sqrt[x] / Sqrt[a]] ((a + b x) / a) ^ (5 / 2) + 15 a ^ 6 b ^ (13 / 2) S
qrt[x] (a + b x) ^ 2 + 5 a ^ 5 b ^ (15 / 2) x ^ (3 / 2) (a + b x) ^ 2 - 2 a ^ 4 b ^ (17 / 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) / (1536 a ^ (5 / 2) b ^ (19 / 2) ((a + b x) / a) ^ (5 / 2))

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Maple [A]
time = 0.10, size = 160, normalized size = 0.98

method	result	size
risch	\(\frac {\left (256 b^{5} x^{5}+640 a \,b^{4} x^{4}+432 a^{2} b^{3} x^{3}+8 a^{3} b^{2} x^{2}-10 a^{4} b x +15 a^{5}\right ) \sqrt {x}\, \sqrt {b x +a}}{1536 b^{3}}-\frac {5 a^{6} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right ) \sqrt {x \left (b x +a \right )}}{1024 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +a}}\)	\(120\)
default	\(\frac {x^{\frac {5}{2}} \left (b x +a \right )^{\frac {7}{2}}}{6 b}-\frac {5 a \left (\frac {x^{\frac {3}{2}} \left (b x +a \right )^{\frac {7}{2}}}{5 b}-\frac {3 a \left (\frac {\sqrt {x}\, \left (b x +a \right )^{\frac {7}{2}}}{4 b}-\frac {a \left (\frac {\left (b x +a \right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 a \left (\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )}{12 b}\)	\(160\)

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)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))/b^(1/2))))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (118) = 236\).
time = 0.37, size = 244, normalized size = 1.49 \begin {gather*} \frac {5 \, a^{6} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{1024 \, 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]

5/1024*a^6*log(-(sqrt(b) - sqrt(b*x + a)/sqrt(x))/(sqrt(b) + sqrt(b*x + a)/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 = 206, normalized size = 1.26 \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}}{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 + 43
2*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*sqrt(-b)*a
rctan(sqrt(b*x + a)*sqrt(-b)/(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 - 1
0*a^4*b^2*x + 15*a^5*b)*sqrt(b*x + a)*sqrt(x))/b^4]

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Sympy [A]
time = 179.38, size = 207, normalized size = 1.26 \begin {gather*} \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 {asinh}{\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}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(512*b**(7
/2)) + b**3*x**(13/2)/(6*sqrt(a)*sqrt(1 + b*x/a))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (118) = 236\).
time = 0.02, size = 546, normalized size = 3.33 \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)))/b^(7/2))*a^2 + 1/960*((2*(4*(6*(8*x + a/b)*x - 7*a^2/b^2)*x + 35*a^3/b^3)*x - 105*a^4/b^4)
*sqrt(b*x + a)*sqrt(x) - 105*a^5*log(abs(-sqrt(b)*sqrt(x) + sqrt(b*x + a)))/b^(9/2))*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)))/b^(11/2))*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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