Integrand size = 33, antiderivative size = 347 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=-\frac {20 b^3 (b d-a e)^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac {2 b^5 (b d-a e) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {b^6 (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}+\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
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Time = 0.19 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{2 e^7 (a+b x) (d+e x)^2}+\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^7 (a+b x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{e^7 (a+b x)}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}{2 e^7 (a+b x)}-\frac {20 b^3 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x)} \]
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Rule 21
Rule 45
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 )^5}{(d+e x)^3} \, dx}{b^4 \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)^6}{(d+e x)^3} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (-\frac {20 b^3 (b d-a e)^3}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^3}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^2}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^2}{e^6}+\frac {b^6 (d+e x)^3}{e^6}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {20 b^3 (b d-a e)^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac {2 b^5 (b d-a e) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {b^6 (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}+\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \\ \end{align*}
Time = 1.10 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {\sqrt {(a+b x)^2} \left (-2 a^6 e^6-12 a^5 b e^5 (d+2 e x)+30 a^4 b^2 d e^4 (3 d+4 e x)+40 a^3 b^3 e^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+30 a^2 b^4 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+4 a b^5 e \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+b^6 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )+60 b^2 (b d-a e)^4 (d+e x)^2 \log (d+e x)\right )}{4 e^7 (a+b x) (d+e x)^2} \]
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Time = 0.44 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.15
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} \left (\frac {1}{4} b^{3} x^{4} e^{3}+2 x^{3} a \,b^{2} e^{3}-x^{3} b^{3} d \,e^{2}+\frac {15}{2} x^{2} a^{2} b \,e^{3}-9 x^{2} a \,b^{2} d \,e^{2}+3 x^{2} b^{3} d^{2} e +20 a^{3} e^{3} x -45 a^{2} b d \,e^{2} x +36 a \,b^{2} d^{2} e x -10 b^{3} d^{3} x \right )}{\left (b x +a \right ) e^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-6 e^{5} a^{5} b +30 d \,e^{4} a^{4} b^{2}-60 d^{2} e^{3} a^{3} b^{3}+60 d^{3} e^{2} b^{4} a^{2}-30 d^{4} e \,b^{5} a +6 d^{5} b^{6}\right ) x -\frac {e^{6} a^{6}+6 b d \,e^{5} a^{5}-45 b^{2} d^{2} e^{4} a^{4}+100 b^{3} d^{3} e^{3} a^{3}-105 b^{4} d^{4} e^{2} a^{2}+54 b^{5} d^{5} e a -11 b^{6} d^{6}}{2 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{2}}+\frac {15 \sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\) | \(398\) |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (90 b^{2} d^{2} e^{4} a^{4}-200 b^{3} d^{3} e^{3} a^{3}+210 b^{4} d^{4} e^{2} a^{2}-108 b^{5} d^{5} e a -12 b d \,e^{5} a^{5}-24 a^{5} b \,e^{6} x -16 b^{6} d^{5} e x +8 a \,b^{5} e^{6} x^{5}-2 b^{6} d \,e^{5} x^{5}+30 a^{2} b^{4} e^{6} x^{4}+5 b^{6} d^{2} e^{4} x^{4}+80 a^{3} b^{3} e^{6} x^{3}-20 b^{6} d^{3} e^{3} x^{3}-68 b^{6} d^{4} e^{2} x^{2}-20 a \,b^{5} d \,e^{5} x^{4}-120 a^{2} b^{4} d \,e^{5} x^{3}+80 a \,b^{5} d^{2} e^{4} x^{3}+160 a^{3} b^{3} d \,e^{5} x^{2}-330 a^{2} b^{4} d^{2} e^{4} x^{2}+252 a \,b^{5} d^{3} e^{3} x^{2}+120 a^{4} b^{2} d \,e^{5} x -240 \ln \left (e x +d \right ) a^{3} b^{3} d \,e^{5} x^{2}+360 \ln \left (e x +d \right ) a^{2} b^{4} d^{2} e^{4} x^{2}-240 \ln \left (e x +d \right ) a \,b^{5} d^{3} e^{3} x^{2}+120 \ln \left (e x +d \right ) a^{4} b^{2} d \,e^{5} x -480 \ln \left (e x +d \right ) a^{3} b^{3} d^{2} e^{4} x +720 \ln \left (e x +d \right ) a^{2} b^{4} d^{3} e^{3} x -480 \ln \left (e x +d \right ) a \,b^{5} d^{4} e^{2} x +60 \ln \left (e x +d \right ) a^{4} b^{2} e^{6} x^{2}+60 \ln \left (e x +d \right ) b^{6} d^{4} e^{2} x^{2}-2 e^{6} a^{6}+22 b^{6} d^{6}-240 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}+360 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}-240 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +60 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}-160 a^{3} b^{3} d^{2} e^{4} x +60 a^{2} b^{4} d^{3} e^{3} x +24 a \,b^{5} d^{4} e^{2} x +120 \ln \left (e x +d \right ) b^{6} d^{5} e x +b^{6} e^{6} x^{6}+60 \ln \left (e x +d \right ) b^{6} d^{6}\right )}{4 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )^{2}}\) | \(669\) |
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Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (264) = 528\).
Time = 0.30 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {b^{6} e^{6} x^{6} + 22 \, b^{6} d^{6} - 108 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} d^{4} e^{2} - 200 \, a^{3} b^{3} d^{3} e^{3} + 90 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 2 \, {\left (b^{6} d e^{5} - 4 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} - 4 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} - 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 2 \, {\left (34 \, b^{6} d^{4} e^{2} - 126 \, a b^{5} d^{3} e^{3} + 165 \, a^{2} b^{4} d^{2} e^{4} - 80 \, a^{3} b^{3} d e^{5}\right )} x^{2} - 4 \, {\left (4 \, b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} - 15 \, a^{2} b^{4} d^{3} e^{3} + 40 \, a^{3} b^{3} d^{2} e^{4} - 30 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 4 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} - 4 \, a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 2 \, {\left (b^{6} d^{5} e - 4 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} - 4 \, a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]
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\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (264) = 528\).
Time = 0.28 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {15 \, {\left (b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {11 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 54 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 105 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 100 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 12 \, {\left (b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{7}} + \frac {b^{6} e^{9} x^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, b^{6} d e^{8} x^{3} \mathrm {sgn}\left (b x + a\right ) + 8 \, a b^{5} e^{9} x^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, b^{6} d^{2} e^{7} x^{2} \mathrm {sgn}\left (b x + a\right ) - 36 \, a b^{5} d e^{8} x^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} e^{9} x^{2} \mathrm {sgn}\left (b x + a\right ) - 40 \, b^{6} d^{3} e^{6} x \mathrm {sgn}\left (b x + a\right ) + 144 \, a b^{5} d^{2} e^{7} x \mathrm {sgn}\left (b x + a\right ) - 180 \, a^{2} b^{4} d e^{8} x \mathrm {sgn}\left (b x + a\right ) + 80 \, a^{3} b^{3} e^{9} x \mathrm {sgn}\left (b x + a\right )}{4 \, e^{12}} \]
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Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \]
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