Optimal. Leaf size=181 \[ \frac {a^4 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^5}-\frac {4 a^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {3 a^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^5}-\frac {4 a (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac {(a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^5} \]
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Rubi [A]
time = 0.04, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {659}
\begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}-\frac {4 a \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5}+\frac {3 a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{4 b^5}+\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^5}-\frac {4 a^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 659
Rubi steps
\begin {align*} \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^4 \left (a b+b^2 x\right )^5}{b^4}-\frac {4 a^3 \left (a b+b^2 x\right )^6}{b^5}+\frac {6 a^2 \left (a b+b^2 x\right )^7}{b^6}-\frac {4 a \left (a b+b^2 x\right )^8}{b^7}+\frac {\left (a b+b^2 x\right )^9}{b^8}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {a^4 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^5}-\frac {4 a^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {3 a^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^5}-\frac {4 a (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac {(a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 77, normalized size = 0.43 \begin {gather*} \frac {x^5 \sqrt {(a+b x)^2} \left (252 a^5+1050 a^4 b x+1800 a^3 b^2 x^2+1575 a^2 b^3 x^3+700 a b^4 x^4+126 b^5 x^5\right )}{1260 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 74, normalized size = 0.41
method | result | size |
gosper | \(\frac {x^{5} \left (126 b^{5} x^{5}+700 a \,b^{4} x^{4}+1575 a^{2} b^{3} x^{3}+1800 a^{3} x^{2} b^{2}+1050 a^{4} b x +252 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{1260 \left (b x +a \right )^{5}}\) | \(74\) |
default | \(\frac {x^{5} \left (126 b^{5} x^{5}+700 a \,b^{4} x^{4}+1575 a^{2} b^{3} x^{3}+1800 a^{3} x^{2} b^{2}+1050 a^{4} b x +252 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{1260 \left (b x +a \right )^{5}}\) | \(74\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{5} x^{10}}{10 b x +10 a}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, a \,b^{4} x^{9}}{9 \left (b x +a \right )}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, a^{2} b^{3} x^{8}}{4 \left (b x +a \right )}+\frac {10 \sqrt {\left (b x +a \right )^{2}}\, a^{3} b^{2} x^{7}}{7 \left (b x +a \right )}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, a^{4} b \,x^{6}}{6 \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{5} x^{5}}{5 b x +5 a}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 160, normalized size = 0.88 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{3}}{10 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} x}{6 \, b^{4}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x^{2}}{90 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5}}{6 \, b^{5}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} x}{180 \, b^{4}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3}}{1260 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.62, size = 57, normalized size = 0.31 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} + \frac {5}{9} \, a b^{4} x^{9} + \frac {5}{4} \, a^{2} b^{3} x^{8} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{5} \, a^{5} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.69, size = 107, normalized size = 0.59 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{9} \, a b^{4} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{2} b^{3} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, a^{3} b^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, a^{4} b x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{10} \mathrm {sgn}\left (b x + a\right )}{1260 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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