Optimal. Leaf size=158 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{5 e^3 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{6 e^3 (a+b x) (d+e x)^6}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4} \]
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Rubi [A] time = 0.10, 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)}{5 e^3 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{6 e^3 (a+b x) (d+e x)^6}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4} \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)^7} \, 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)^7} \, 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)^7}+\frac {b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^6}+\frac {b^2 B}{e^2 (d+e x)^5}\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}}{6 e^3 (a+b x) (d+e x)^6}+\frac {(2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 82, normalized size = 0.52 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (2 a e (5 A e+B (d+6 e x))+b \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )\right )}{60 e^3 (a+b x) (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.55, 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 = 126, normalized size = 0.80 \begin {gather*} -\frac {15 \, B b e^{2} x^{2} + B b d^{2} + 10 \, A a e^{2} + 2 \, {\left (B a + A b\right )} d e + 6 \, {\left (B b d e + 2 \, {\left (B a + A b\right )} e^{2}\right )} x}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} 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 = 118, normalized size = 0.75 \begin {gather*} -\frac {{\left (15 \, B b x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, B b d x e \mathrm {sgn}\left (b x + a\right ) + B b d^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, B a x e^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, A b x e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a d e \mathrm {sgn}\left (b x + a\right ) + 2 \, A b d e \mathrm {sgn}\left (b x + a\right ) + 10 \, A a e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 88, normalized size = 0.56 \begin {gather*} -\frac {\left (15 B b \,e^{2} x^{2}+12 A b \,e^{2} x +12 B a \,e^{2} x +6 B b d e x +10 A a \,e^{2}+2 A b d e +2 B a d e +B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{60 \left (e x +d \right )^{6} \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 = 2.16, size = 87, normalized size = 0.55 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (10\,A\,a\,e^2+B\,b\,d^2+12\,A\,b\,e^2\,x+12\,B\,a\,e^2\,x+15\,B\,b\,e^2\,x^2+2\,A\,b\,d\,e+2\,B\,a\,d\,e+6\,B\,b\,d\,e\,x\right )}{60\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.57, size = 144, normalized size = 0.91 \begin {gather*} \frac {- 10 A a e^{2} - 2 A b d e - 2 B a d e - B b d^{2} - 15 B b e^{2} x^{2} + x \left (- 12 A b e^{2} - 12 B a e^{2} - 6 B b d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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