Integrand size = 28, antiderivative size = 222 \[ \int \frac {(d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {e (b d-a e)^3 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e)^2 (a+b x) (d+e x)^2}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e) (a+b x) (d+e x)^3}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^4}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e)^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int \frac {(d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {(a+b x) (d+e x)^3 (b d-a e)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^4}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (b d-a e)^4 \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x (a+b x) (b d-a e)^3}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^2 (b d-a e)^2}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {(d+e x)^4}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (a b+b^2 x\right ) \int \left (\frac {e (b d-a e)^3}{b^5}+\frac {(b d-a e)^4}{b^4 \left (a b+b^2 x\right )}+\frac {e (b d-a e)^2 (d+e x)}{b^4}+\frac {e (b d-a e) (d+e x)^2}{b^3}+\frac {e (d+e x)^3}{b^2}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {e (b d-a e)^3 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e)^2 (a+b x) (d+e x)^2}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e) (a+b x) (d+e x)^3}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^4}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e)^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.59 \[ \int \frac {(d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {(a+b x) \left (b e x \left (-12 a^3 e^3+6 a^2 b e^2 (8 d+e x)-4 a b^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+b^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (a+b x)\right )}{12 b^5 \sqrt {(a+b x)^2}} \]
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Time = 2.42 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.99
method | result | size |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}\, e \left (-\frac {b^{3} x^{4} e^{3}}{4}+\frac {\left (\left (a e -2 b d \right ) b^{2} e^{2}-2 b^{3} d \,e^{2}\right ) x^{3}}{3}+\frac {\left (2 \left (a e -2 b d \right ) b^{2} d e -b e \left (a^{2} e^{2}-2 a b d e +2 b^{2} d^{2}\right )\right ) x^{2}}{2}+\left (a e -2 b d \right ) \left (a^{2} e^{2}-2 a b d e +2 b^{2} d^{2}\right ) x \right )}{\left (b x +a \right ) b^{4}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{5}}\) | \(220\) |
default | \(\frac {\left (b x +a \right ) \left (3 b^{4} x^{4} e^{4}-4 x^{3} a \,b^{3} e^{4}+16 x^{3} b^{4} d \,e^{3}+6 x^{2} a^{2} b^{2} e^{4}-24 x^{2} a \,b^{3} d \,e^{3}+36 x^{2} b^{4} d^{2} e^{2}+12 \ln \left (b x +a \right ) a^{4} e^{4}-48 \ln \left (b x +a \right ) a^{3} b d \,e^{3}+72 \ln \left (b x +a \right ) a^{2} b^{2} d^{2} e^{2}-48 \ln \left (b x +a \right ) a \,b^{3} d^{3} e +12 \ln \left (b x +a \right ) b^{4} d^{4}-12 x \,a^{3} b \,e^{4}+48 x \,a^{2} b^{2} d \,e^{3}-72 x a \,b^{3} d^{2} e^{2}+48 x \,b^{4} d^{3} e \right )}{12 \sqrt {\left (b x +a \right )^{2}}\, b^{5}}\) | \(223\) |
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Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {3 \, b^{4} e^{4} x^{4} + 4 \, {\left (4 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (6 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 12 \, {\left (4 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 4 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (162) = 324\).
Time = 1.54 (sec) , antiderivative size = 746, normalized size of antiderivative = 3.36 \[ \int \frac {(d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {e^{4} x^{3}}{4 b^{2}} + \frac {x^{2} \left (- \frac {7 a e^{4}}{4 b} + 4 d e^{3}\right )}{3 b^{2}} + \frac {x \left (- \frac {3 a^{2} e^{4}}{4 b^{2}} - \frac {5 a \left (- \frac {7 a e^{4}}{4 b} + 4 d e^{3}\right )}{3 b} + 6 d^{2} e^{2}\right )}{2 b^{2}} + \frac {- \frac {2 a^{2} \left (- \frac {7 a e^{4}}{4 b} + 4 d e^{3}\right )}{3 b^{2}} - \frac {3 a \left (- \frac {3 a^{2} e^{4}}{4 b^{2}} - \frac {5 a \left (- \frac {7 a e^{4}}{4 b} + 4 d e^{3}\right )}{3 b} + 6 d^{2} e^{2}\right )}{2 b} + 4 d^{3} e}{b^{2}}\right ) + \frac {\left (\frac {a}{b} + x\right ) \left (- \frac {a^{2} \left (- \frac {3 a^{2} e^{4}}{4 b^{2}} - \frac {5 a \left (- \frac {7 a e^{4}}{4 b} + 4 d e^{3}\right )}{3 b} + 6 d^{2} e^{2}\right )}{2 b^{2}} - \frac {a \left (- \frac {2 a^{2} \left (- \frac {7 a e^{4}}{4 b} + 4 d e^{3}\right )}{3 b^{2}} - \frac {3 a \left (- \frac {3 a^{2} e^{4}}{4 b^{2}} - \frac {5 a \left (- \frac {7 a e^{4}}{4 b} + 4 d e^{3}\right )}{3 b} + 6 d^{2} e^{2}\right )}{2 b} + 4 d^{3} e\right )}{b} + d^{4}\right ) \log {\left (\frac {a}{b} + x \right )}}{\sqrt {b^{2} \left (\frac {a}{b} + x\right )^{2}}} & \text {for}\: b^{2} \neq 0 \\\frac {2 d^{4} \sqrt {a^{2} + 2 a b x} + \frac {4 d^{3} e \left (- a^{2} \sqrt {a^{2} + 2 a b x} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3}\right )}{a b} + \frac {3 d^{2} e^{2} \left (a^{4} \sqrt {a^{2} + 2 a b x} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}\right )}{a^{2} b^{2}} + \frac {d e^{3} \left (- a^{6} \sqrt {a^{2} + 2 a b x} + a^{4} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}} - \frac {3 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7}\right )}{a^{3} b^{3}} + \frac {e^{4} \left (a^{8} \sqrt {a^{2} + 2 a b x} - \frac {4 a^{6} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {6 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} - \frac {4 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9}\right )}{8 a^{4} b^{4}}}{2 a b} & \text {for}\: a b \neq 0 \\\frac {d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac {e^{4} x^{5}}{5}}{\sqrt {a^{2}}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (161) = 322\).
Time = 0.20 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.57 \[ \int \frac {(d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} e^{4} x^{3}}{4 \, b^{2}} + \frac {3 \, d^{2} e^{2} x^{2}}{b} - \frac {10 \, a d e^{3} x^{2}}{3 \, b^{2}} + \frac {13 \, a^{2} e^{4} x^{2}}{12 \, b^{3}} + \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} d e^{3} x^{2}}{3 \, b^{2}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{4} x^{2}}{12 \, b^{3}} - \frac {6 \, a d^{2} e^{2} x}{b^{2}} + \frac {20 \, a^{2} d e^{3} x}{3 \, b^{3}} - \frac {13 \, a^{3} e^{4} x}{6 \, b^{4}} + \frac {d^{4} \log \left (x + \frac {a}{b}\right )}{b} - \frac {4 \, a d^{3} e \log \left (x + \frac {a}{b}\right )}{b^{2}} + \frac {6 \, a^{2} d^{2} e^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {4 \, a^{3} d e^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {a^{4} e^{4} \log \left (x + \frac {a}{b}\right )}{b^{5}} + \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} d^{3} e}{b^{2}} - \frac {8 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d e^{3}}{3 \, b^{4}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{4}}{6 \, b^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {3 \, b^{3} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 16 \, b^{3} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, b^{3} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) - 24 \, a b^{2} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 48 \, b^{3} d^{3} e x \mathrm {sgn}\left (b x + a\right ) - 72 \, a b^{2} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 48 \, a^{2} b d e^{3} x \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{3} e^{4} x \mathrm {sgn}\left (b x + a\right )}{12 \, b^{4}} + \frac {{\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \]
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Timed out. \[ \int \frac {(d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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