Optimal. Leaf size=158 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{4 e^3 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{5 e^3 (a+b x) (d+e x)^5}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3} \]
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Rubi [A] time = 0.09, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{4 e^3 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{5 e^3 (a+b x) (d+e x)^5}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^6} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right ) (A+B x)}{(d+e x)^6} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (-B d+A e)}{e^2 (d+e x)^6}+\frac {b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^5}+\frac {b^2 B}{e^2 (d+e x)^4}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e) (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}+\frac {(2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 83, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3 a e (4 A e+B (d+5 e x))+b \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )\right )}{60 e^3 (a+b x) (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.42, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 117, normalized size = 0.74 \begin {gather*} -\frac {20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \, {\left (B a + A b\right )} d e + 5 \, {\left (2 \, B b d e + 3 \, {\left (B a + A b\right )} e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 119, normalized size = 0.75 \begin {gather*} -\frac {{\left (20 \, B b x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, B b d x e \mathrm {sgn}\left (b x + a\right ) + 2 \, B b d^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, B a x e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, A b x e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a d e \mathrm {sgn}\left (b x + a\right ) + 3 \, A b d e \mathrm {sgn}\left (b x + a\right ) + 12 \, A a e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{60 \, {\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 89, normalized size = 0.56 \begin {gather*} -\frac {\left (20 B b \,e^{2} x^{2}+15 A b \,e^{2} x +15 B a \,e^{2} x +10 B b d e x +12 A a \,e^{2}+3 A b d e +3 B a d e +2 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{60 \left (e x +d \right )^{5} \left (b x +a \right ) e^{3}} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.17, size = 88, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (12\,A\,a\,e^2+2\,B\,b\,d^2+15\,A\,b\,e^2\,x+15\,B\,a\,e^2\,x+20\,B\,b\,e^2\,x^2+3\,A\,b\,d\,e+3\,B\,a\,d\,e+10\,B\,b\,d\,e\,x\right )}{60\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.36, size = 134, normalized size = 0.85 \begin {gather*} \frac {- 12 A a e^{2} - 3 A b d e - 3 B a d e - 2 B b d^{2} - 20 B b e^{2} x^{2} + x \left (- 15 A b e^{2} - 15 B a e^{2} - 10 B b d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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