Integrand size = 33, antiderivative size = 332 \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (A b-a B)}{2 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (B d-A e) (a+b x)}{2 (b d-a e)^3 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (2 b B d-3 A b e+a B e) (a+b x)}{(b d-a e)^4 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b e (b B d-2 A b e+a B e) (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b e (b B d-2 A b e+a B e) (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {784, 78} \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {b (A b-a B)}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {b (2 a B e-3 A b e+b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {e (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac {e (a+b x) (B d-A e)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}-\frac {3 b e (a+b x) \log (a+b x) (a B e-2 A b e+b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {3 b e (a+b x) \log (d+e x) (a B e-2 A b e+b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5} \]
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Rule 78
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{\left (a b+b^2 x\right )^3 (d+e x)^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {A b-a B}{b (b d-a e)^3 (a+b x)^3}+\frac {b B d-3 A b e+2 a B e}{b (b d-a e)^4 (a+b x)^2}+\frac {3 e (-b B d+2 A b e-a B e)}{b (b d-a e)^5 (a+b x)}-\frac {e^2 (-B d+A e)}{b^3 (b d-a e)^3 (d+e x)^3}-\frac {e^2 (-2 b B d+3 A b e-a B e)}{b^3 (b d-a e)^4 (d+e x)^2}-\frac {3 e^2 (-b B d+2 A b e-a B e)}{b^2 (b d-a e)^5 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (A b-a B)}{2 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (B d-A e) (a+b x)}{2 (b d-a e)^3 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (2 b B d-3 A b e+a B e) (a+b x)}{(b d-a e)^4 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b e (b B d-2 A b e+a B e) (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b e (b B d-2 A b e+a B e) (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.13 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {(a+b x) \left (-b (A b-a B) (b d-a e)^2-2 b (b d-a e) (b B d-3 A b e+2 a B e) (a+b x)+\frac {e (b d-a e)^2 (-B d+A e) (a+b x)^2}{(d+e x)^2}+\frac {2 e (b d-a e) (-2 b B d+3 A b e-a B e) (a+b x)^2}{d+e x}-6 b e (b B d-2 A b e+a B e) (a+b x)^2 \log (a+b x)+6 b e (b B d-2 A b e+a B e) (a+b x)^2 \log (d+e x)\right )}{2 (b d-a e)^5 \left ((a+b x)^2\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(650\) vs. \(2(262)=524\).
Time = 0.45 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.96
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {3 b^{2} e^{2} \left (2 A b e -B a e -B b d \right ) x^{3}}{e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}+\frac {9 b e \left (a e +b d \right ) \left (2 A b e -B a e -B b d \right ) x^{2}}{2 \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}+\frac {\left (2 A \,a^{2} b \,e^{3}+14 A a \,b^{2} d \,e^{2}+2 A \,b^{3} d^{2} e -B \,e^{3} a^{3}-8 B \,a^{2} b d \,e^{2}-8 B a \,b^{2} d^{2} e -B \,b^{3} d^{3}\right ) x}{e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}-\frac {A \,a^{3} e^{3}-7 A \,a^{2} b d \,e^{2}-7 A a \,b^{2} d^{2} e +A \,b^{3} d^{3}+B \,a^{3} d \,e^{2}+10 B \,a^{2} b \,d^{2} e +B a \,b^{2} d^{3}}{2 \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}\right )}{\left (b x +a \right )^{3} \left (e x +d \right )^{2}}+\frac {3 \sqrt {\left (b x +a \right )^{2}}\, b e \left (2 A b e -B a e -B b d \right ) \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}-\frac {3 \sqrt {\left (b x +a \right )^{2}}\, b e \left (2 A b e -B a e -B b d \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}\) | \(651\) |
default | \(\text {Expression too large to display}\) | \(1271\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1215 vs. \(2 (262) = 524\).
Time = 0.40 (sec) , antiderivative size = 1215, normalized size of antiderivative = 3.66 \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 634 vs. \(2 (262) = 524\).
Time = 0.30 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.91 \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {3 \, {\left (B b^{3} d e + B a b^{2} e^{2} - 2 \, A b^{3} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {3 \, {\left (B b^{2} d e^{2} + B a b e^{3} - 2 \, A b^{2} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {6 \, B b^{3} d e^{2} x^{3} + 6 \, B a b^{2} e^{3} x^{3} - 12 \, A b^{3} e^{3} x^{3} + 9 \, B b^{3} d^{2} e x^{2} + 18 \, B a b^{2} d e^{2} x^{2} - 18 \, A b^{3} d e^{2} x^{2} + 9 \, B a^{2} b e^{3} x^{2} - 18 \, A a b^{2} e^{3} x^{2} + 2 \, B b^{3} d^{3} x + 16 \, B a b^{2} d^{2} e x - 4 \, A b^{3} d^{2} e x + 16 \, B a^{2} b d e^{2} x - 28 \, A a b^{2} d e^{2} x + 2 \, B a^{3} e^{3} x - 4 \, A a^{2} b e^{3} x + B a b^{2} d^{3} + A b^{3} d^{3} + 10 \, B a^{2} b d^{2} e - 7 \, A a b^{2} d^{2} e + B a^{3} d e^{2} - 7 \, A a^{2} b d e^{2} + A a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} {\left (b e x^{2} + b d x + a e x + a d\right )}^{2}} \]
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Timed out. \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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