Integrand size = 33, antiderivative size = 98 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (b d-a e) (d+e x)^6}+\frac {b (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^5} \]
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Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {784, 21, 47, 37} \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4}{30 (d+e x)^5 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4}{6 (d+e x)^6 (b d-a e)} \]
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Rule 21
Rule 37
Rule 47
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^7} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^7} \, dx}{a b+b^2 x} \\ & = \frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (b d-a e) (d+e x)^6}+\frac {\left (b^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^6} \, dx}{6 (b d-a e) \left (a b+b^2 x\right )} \\ & = \frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (b d-a e) (d+e x)^6}+\frac {b (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^5} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.65 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {\sqrt {(a+b x)^2} \left (5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b^3 e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )}{30 e^5 (a+b x) (d+e x)^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(186\) vs. \(2(72)=144\).
Time = 0.94 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.91
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{4} x^{4}}{2 e}-\frac {2 b^{3} \left (2 a e +b d \right ) x^{3}}{3 e^{2}}-\frac {b^{2} \left (3 e^{2} a^{2}+2 a b d e +b^{2} d^{2}\right ) x^{2}}{2 e^{3}}-\frac {b \left (4 a^{3} e^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{5 e^{4}}-\frac {5 e^{4} a^{4}+4 b d \,e^{3} a^{3}+3 b^{2} d^{2} e^{2} a^{2}+2 b^{3} d^{3} e a +b^{4} d^{4}}{30 e^{5}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}\) | \(187\) |
gosper | \(-\frac {\left (15 e^{4} x^{4} b^{4}+40 x^{3} a \,b^{3} e^{4}+20 x^{3} b^{4} d \,e^{3}+45 x^{2} a^{2} b^{2} e^{4}+30 x^{2} a \,b^{3} d \,e^{3}+15 x^{2} b^{4} d^{2} e^{2}+24 x \,a^{3} b \,e^{4}+18 x \,a^{2} b^{2} d \,e^{3}+12 x a \,b^{3} d^{2} e^{2}+6 x \,b^{4} d^{3} e +5 e^{4} a^{4}+4 b d \,e^{3} a^{3}+3 b^{2} d^{2} e^{2} a^{2}+2 b^{3} d^{3} e a +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{30 e^{5} \left (e x +d \right )^{6} \left (b x +a \right )^{3}}\) | \(201\) |
default | \(-\frac {\left (15 e^{4} x^{4} b^{4}+40 x^{3} a \,b^{3} e^{4}+20 x^{3} b^{4} d \,e^{3}+45 x^{2} a^{2} b^{2} e^{4}+30 x^{2} a \,b^{3} d \,e^{3}+15 x^{2} b^{4} d^{2} e^{2}+24 x \,a^{3} b \,e^{4}+18 x \,a^{2} b^{2} d \,e^{3}+12 x a \,b^{3} d^{2} e^{2}+6 x \,b^{4} d^{3} e +5 e^{4} a^{4}+4 b d \,e^{3} a^{3}+3 b^{2} d^{2} e^{2} a^{2}+2 b^{3} d^{3} e a +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{30 e^{5} \left (e x +d \right )^{6} \left (b x +a \right )^{3}}\) | \(201\) |
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (72) = 144\).
Time = 0.30 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {15 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4} + 20 \, {\left (b^{4} d e^{3} + 2 \, a b^{3} e^{4}\right )} x^{3} + 15 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 2 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + 4 \, a^{3} b e^{4}\right )} x}{30 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]
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\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{7}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (72) = 144\).
Time = 0.27 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.20 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {b^{6} \mathrm {sgn}\left (b x + a\right )}{30 \, {\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )}} - \frac {15 \, b^{4} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, b^{4} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 40 \, a b^{3} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{4} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, a b^{3} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{2} b^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{4} d^{3} e x \mathrm {sgn}\left (b x + a\right ) + 12 \, a b^{3} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{2} b^{2} d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 24 \, a^{3} b e^{4} x \mathrm {sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )}{30 \, {\left (e x + d\right )}^{6} e^{5}} \]
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Time = 10.80 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.58 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {\left (\frac {-4\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-4\,a\,b^3\,d^2\,e+b^4\,d^3}{5\,e^5}+\frac {d\,\left (\frac {d\,\left (\frac {b^4\,d}{5\,e^3}-\frac {b^3\,\left (4\,a\,e-b\,d\right )}{5\,e^3}\right )}{e}+\frac {b^2\,\left (6\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{5\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {a^4}{6\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {2\,a\,b^3}{3\,e}-\frac {b^4\,d}{6\,e^2}\right )}{e}-\frac {a^2\,b^2}{e}\right )}{e}+\frac {2\,a^3\,b}{3\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {6\,a^2\,b^2\,e^2-8\,a\,b^3\,d\,e+3\,b^4\,d^2}{4\,e^5}+\frac {d\,\left (\frac {b^4\,d}{4\,e^4}-\frac {b^3\,\left (2\,a\,e-b\,d\right )}{2\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}+\frac {\left (\frac {3\,b^4\,d-4\,a\,b^3\,e}{3\,e^5}+\frac {b^4\,d}{3\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2} \]
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