Optimal. Leaf size=323 \[ -\frac {4 b e^3 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}-\frac {e^3 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^4}-\frac {10 b^2 e^3 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac {10 b^2 e^3 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac {6 b^2 e^2}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {3 b^2 e}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {b^2}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
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Rubi [A] time = 0.24, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 44} \[ -\frac {4 b e^3 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}-\frac {e^3 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^4}-\frac {6 b^2 e^2}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {10 b^2 e^3 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac {10 b^2 e^3 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac {3 b^2 e}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {b^2}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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Rule 21
Rule 44
Rule 770
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {a+b x}{\left (a b+b^2 x\right )^5 (d+e x)^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{(a+b x)^4 (d+e x)^3} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {b^3}{(b d-a e)^3 (a+b x)^4}-\frac {3 b^3 e}{(b d-a e)^4 (a+b x)^3}+\frac {6 b^3 e^2}{(b d-a e)^5 (a+b x)^2}-\frac {10 b^3 e^3}{(b d-a e)^6 (a+b x)}+\frac {e^4}{(b d-a e)^4 (d+e x)^3}+\frac {4 b e^4}{(b d-a e)^5 (d+e x)^2}+\frac {10 b^2 e^4}{(b d-a e)^6 (d+e x)}\right ) \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {6 b^2 e^2}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{3 (b d-a e)^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b^2 e}{2 (b d-a e)^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (a+b x)}{2 (b d-a e)^4 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 b e^3 (a+b x)}{(b d-a e)^5 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {10 b^2 e^3 (a+b x) \log (a+b x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {10 b^2 e^3 (a+b x) \log (d+e x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 184, normalized size = 0.57 \[ \frac {60 b^2 e^3 (a+b x)^3 \log (d+e x)-36 b^2 e^2 (a+b x)^2 (b d-a e)+9 b^2 e (a+b x) (b d-a e)^2-2 b^2 (b d-a e)^3-60 b^2 e^3 (a+b x)^3 \log (a+b x)-\frac {3 e^3 (a+b x)^3 (b d-a e)^2}{(d+e x)^2}-\frac {24 b e^3 (a+b x)^3 (b d-a e)}{d+e x}}{6 \left ((a+b x)^2\right )^{3/2} (b d-a e)^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 1151, normalized size = 3.56 \[ -\frac {2 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 30 \, a^{4} b d e^{4} + 3 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 30 \, {\left (3 \, b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 5 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} + 21 \, a b^{4} d^{2} e^{3} - 12 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 24 \, a^{2} b^{3} d^{2} e^{3} + 32 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x + 60 \, {\left (b^{5} e^{5} x^{5} + a^{3} b^{2} d^{2} e^{3} + {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + {\left (3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{3} d^{2} e^{3} + 2 \, a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (b x + a\right ) - 60 \, {\left (b^{5} e^{5} x^{5} + a^{3} b^{2} d^{2} e^{3} + {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + {\left (3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{3} d^{2} e^{3} + 2 \, a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (a^{3} b^{6} d^{8} - 6 \, a^{4} b^{5} d^{7} e + 15 \, a^{5} b^{4} d^{6} e^{2} - 20 \, a^{6} b^{3} d^{5} e^{3} + 15 \, a^{7} b^{2} d^{4} e^{4} - 6 \, a^{8} b d^{3} e^{5} + a^{9} d^{2} e^{6} + {\left (b^{9} d^{6} e^{2} - 6 \, a b^{8} d^{5} e^{3} + 15 \, a^{2} b^{7} d^{4} e^{4} - 20 \, a^{3} b^{6} d^{3} e^{5} + 15 \, a^{4} b^{5} d^{2} e^{6} - 6 \, a^{5} b^{4} d e^{7} + a^{6} b^{3} e^{8}\right )} x^{5} + {\left (2 \, b^{9} d^{7} e - 9 \, a b^{8} d^{6} e^{2} + 12 \, a^{2} b^{7} d^{5} e^{3} + 5 \, a^{3} b^{6} d^{4} e^{4} - 30 \, a^{4} b^{5} d^{3} e^{5} + 33 \, a^{5} b^{4} d^{2} e^{6} - 16 \, a^{6} b^{3} d e^{7} + 3 \, a^{7} b^{2} e^{8}\right )} x^{4} + {\left (b^{9} d^{8} - 18 \, a^{2} b^{7} d^{6} e^{2} + 52 \, a^{3} b^{6} d^{5} e^{3} - 60 \, a^{4} b^{5} d^{4} e^{4} + 24 \, a^{5} b^{4} d^{3} e^{5} + 10 \, a^{6} b^{3} d^{2} e^{6} - 12 \, a^{7} b^{2} d e^{7} + 3 \, a^{8} b e^{8}\right )} x^{3} + {\left (3 \, a b^{8} d^{8} - 12 \, a^{2} b^{7} d^{7} e + 10 \, a^{3} b^{6} d^{6} e^{2} + 24 \, a^{4} b^{5} d^{5} e^{3} - 60 \, a^{5} b^{4} d^{4} e^{4} + 52 \, a^{6} b^{3} d^{3} e^{5} - 18 \, a^{7} b^{2} d^{2} e^{6} + a^{9} e^{8}\right )} x^{2} + {\left (3 \, a^{2} b^{7} d^{8} - 16 \, a^{3} b^{6} d^{7} e + 33 \, a^{4} b^{5} d^{6} e^{2} - 30 \, a^{5} b^{4} d^{5} e^{3} + 5 \, a^{6} b^{3} d^{4} e^{4} + 12 \, a^{7} b^{2} d^{3} e^{5} - 9 \, a^{8} b d^{2} e^{6} + 2 \, a^{9} d e^{7}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 753, normalized size = 2.33 \[ -\frac {\left (60 b^{5} e^{5} x^{5} \ln \left (b x +a \right )-60 b^{5} e^{5} x^{5} \ln \left (e x +d \right )+180 a \,b^{4} e^{5} x^{4} \ln \left (b x +a \right )-180 a \,b^{4} e^{5} x^{4} \ln \left (e x +d \right )+120 b^{5} d \,e^{4} x^{4} \ln \left (b x +a \right )-120 b^{5} d \,e^{4} x^{4} \ln \left (e x +d \right )+180 a^{2} b^{3} e^{5} x^{3} \ln \left (b x +a \right )-180 a^{2} b^{3} e^{5} x^{3} \ln \left (e x +d \right )+360 a \,b^{4} d \,e^{4} x^{3} \ln \left (b x +a \right )-360 a \,b^{4} d \,e^{4} x^{3} \ln \left (e x +d \right )-60 a \,b^{4} e^{5} x^{4}+60 b^{5} d^{2} e^{3} x^{3} \ln \left (b x +a \right )-60 b^{5} d^{2} e^{3} x^{3} \ln \left (e x +d \right )+60 b^{5} d \,e^{4} x^{4}+60 a^{3} b^{2} e^{5} x^{2} \ln \left (b x +a \right )-60 a^{3} b^{2} e^{5} x^{2} \ln \left (e x +d \right )+360 a^{2} b^{3} d \,e^{4} x^{2} \ln \left (b x +a \right )-360 a^{2} b^{3} d \,e^{4} x^{2} \ln \left (e x +d \right )-150 a^{2} b^{3} e^{5} x^{3}+180 a \,b^{4} d^{2} e^{3} x^{2} \ln \left (b x +a \right )-180 a \,b^{4} d^{2} e^{3} x^{2} \ln \left (e x +d \right )+60 a \,b^{4} d \,e^{4} x^{3}+90 b^{5} d^{2} e^{3} x^{3}+120 a^{3} b^{2} d \,e^{4} x \ln \left (b x +a \right )-120 a^{3} b^{2} d \,e^{4} x \ln \left (e x +d \right )-110 a^{3} b^{2} e^{5} x^{2}+180 a^{2} b^{3} d^{2} e^{3} x \ln \left (b x +a \right )-180 a^{2} b^{3} d^{2} e^{3} x \ln \left (e x +d \right )-120 a^{2} b^{3} d \,e^{4} x^{2}+210 a \,b^{4} d^{2} e^{3} x^{2}+20 b^{5} d^{3} e^{2} x^{2}-15 a^{4} b \,e^{5} x +60 a^{3} b^{2} d^{2} e^{3} \ln \left (b x +a \right )-60 a^{3} b^{2} d^{2} e^{3} \ln \left (e x +d \right )-160 a^{3} b^{2} d \,e^{4} x +120 a^{2} b^{3} d^{2} e^{3} x +60 a \,b^{4} d^{3} e^{2} x -5 b^{5} d^{4} e x +3 a^{5} e^{5}-30 a^{4} b d \,e^{4}-20 a^{3} b^{2} d^{2} e^{3}+60 a^{2} b^{3} d^{3} e^{2}-15 a \,b^{4} d^{4} e +2 b^{5} d^{5}\right ) \left (b x +a \right )^{2}}{6 \left (e x +d \right )^{2} \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,x}{{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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