Integrand size = 24, antiderivative size = 173 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\frac {21}{16} a^4 x \sqrt {a^2-b^2 x^2}-\frac {7 a^3 \left (a^2-b^2 x^2\right )^{3/2}}{8 b}-\frac {21 a^2 (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}-\frac {3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac {21 a^6 \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{16 b} \]
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Time = 0.05 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {685, 655, 201, 223, 209} \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=-\frac {21 a^2 (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}-\frac {3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac {21 a^6 \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{16 b}+\frac {21}{16} a^4 x \sqrt {a^2-b^2 x^2}-\frac {7 a^3 \left (a^2-b^2 x^2\right )^{3/2}}{8 b} \]
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Rule 201
Rule 209
Rule 223
Rule 655
Rule 685
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac {1}{2} (3 a) \int (a+b x)^3 \sqrt {a^2-b^2 x^2} \, dx \\ & = -\frac {3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac {1}{10} \left (21 a^2\right ) \int (a+b x)^2 \sqrt {a^2-b^2 x^2} \, dx \\ & = -\frac {21 a^2 (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}-\frac {3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac {1}{8} \left (21 a^3\right ) \int (a+b x) \sqrt {a^2-b^2 x^2} \, dx \\ & = -\frac {7 a^3 \left (a^2-b^2 x^2\right )^{3/2}}{8 b}-\frac {21 a^2 (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}-\frac {3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac {1}{8} \left (21 a^4\right ) \int \sqrt {a^2-b^2 x^2} \, dx \\ & = \frac {21}{16} a^4 x \sqrt {a^2-b^2 x^2}-\frac {7 a^3 \left (a^2-b^2 x^2\right )^{3/2}}{8 b}-\frac {21 a^2 (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}-\frac {3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac {1}{16} \left (21 a^6\right ) \int \frac {1}{\sqrt {a^2-b^2 x^2}} \, dx \\ & = \frac {21}{16} a^4 x \sqrt {a^2-b^2 x^2}-\frac {7 a^3 \left (a^2-b^2 x^2\right )^{3/2}}{8 b}-\frac {21 a^2 (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}-\frac {3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac {1}{16} \left (21 a^6\right ) \text {Subst}\left (\int \frac {1}{1+b^2 x^2} \, dx,x,\frac {x}{\sqrt {a^2-b^2 x^2}}\right ) \\ & = \frac {21}{16} a^4 x \sqrt {a^2-b^2 x^2}-\frac {7 a^3 \left (a^2-b^2 x^2\right )^{3/2}}{8 b}-\frac {21 a^2 (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}-\frac {3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac {(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac {21 a^6 \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{16 b} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.66 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\frac {\sqrt {a^2-b^2 x^2} \left (-448 a^5-75 a^4 b x+256 a^3 b^2 x^2+350 a^2 b^3 x^3+192 a b^4 x^4+40 b^5 x^5\right )-630 a^6 \arctan \left (\frac {b x}{\sqrt {a^2}-\sqrt {a^2-b^2 x^2}}\right )}{240 b} \]
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Time = 2.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.61
method | result | size |
risch | \(-\frac {\left (-40 b^{5} x^{5}-192 a \,b^{4} x^{4}-350 a^{2} b^{3} x^{3}-256 a^{3} b^{2} x^{2}+75 a^{4} b x +448 a^{5}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{240 b}+\frac {21 a^{6} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{16 \sqrt {b^{2}}}\) | \(105\) |
default | \(a^{4} \left (\frac {x \sqrt {-b^{2} x^{2}+a^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )+b^{4} \left (-\frac {x^{3} \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{6 b^{2}}+\frac {a^{2} \left (-\frac {x \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{4 b^{2}}+\frac {a^{2} \left (\frac {x \sqrt {-b^{2} x^{2}+a^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )}{4 b^{2}}\right )}{2 b^{2}}\right )+4 a \,b^{3} \left (-\frac {x^{2} \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{5 b^{2}}-\frac {2 a^{2} \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{15 b^{4}}\right )+6 a^{2} b^{2} \left (-\frac {x \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{4 b^{2}}+\frac {a^{2} \left (\frac {x \sqrt {-b^{2} x^{2}+a^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )}{4 b^{2}}\right )-\frac {4 a^{3} \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{3 b}\) | \(330\) |
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Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.61 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=-\frac {630 \, a^{6} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) - {\left (40 \, b^{5} x^{5} + 192 \, a b^{4} x^{4} + 350 \, a^{2} b^{3} x^{3} + 256 \, a^{3} b^{2} x^{2} - 75 \, a^{4} b x - 448 \, a^{5}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{240 \, b} \]
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Time = 0.55 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.95 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\begin {cases} \frac {21 a^{6} \left (\begin {cases} \frac {\log {\left (- 2 b^{2} x + 2 \sqrt {- b^{2}} \sqrt {a^{2} - b^{2} x^{2}} \right )}}{\sqrt {- b^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- b^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16} + \sqrt {a^{2} - b^{2} x^{2}} \left (- \frac {28 a^{5}}{15 b} - \frac {5 a^{4} x}{16} + \frac {16 a^{3} b x^{2}}{15} + \frac {35 a^{2} b^{2} x^{3}}{24} + \frac {4 a b^{3} x^{4}}{5} + \frac {b^{4} x^{5}}{6}\right ) & \text {for}\: b^{2} \neq 0 \\\sqrt {a^{2}} \left (\begin {cases} a^{4} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{5}}{5 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.70 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=-\frac {1}{6} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} b^{2} x^{3} + \frac {21 \, a^{6} \arcsin \left (\frac {b x}{a}\right )}{16 \, b} + \frac {21}{16} \, \sqrt {-b^{2} x^{2} + a^{2}} a^{4} x - \frac {4}{5} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} a b x^{2} - \frac {13}{8} \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} a^{2} x - \frac {28 \, {\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}} a^{3}}{15 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.53 \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\frac {21 \, a^{6} \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (b\right )}{16 \, {\left | b \right |}} - \frac {1}{240} \, {\left (\frac {448 \, a^{5}}{b} + {\left (75 \, a^{4} - 2 \, {\left (128 \, a^{3} b + {\left (175 \, a^{2} b^{2} + 4 \, {\left (5 \, b^{4} x + 24 \, a b^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-b^{2} x^{2} + a^{2}} \]
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Timed out. \[ \int (a+b x)^4 \sqrt {a^2-b^2 x^2} \, dx=\int \sqrt {a^2-b^2\,x^2}\,{\left (a+b\,x\right )}^4 \,d x \]
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