Optimal. Leaf size=186 \[ -\frac {(b d-a e) (-3 a B e+2 A b e+b B d)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^2}{2 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (a+b x) \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^2 x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 186, 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} \[ -\frac {(b d-a e) (-3 a B e+2 A b e+b B d)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^2}{2 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (a+b x) \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^2 x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^2}{\left (a b+b^2 x\right )^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 {B e^2}{b^6}+\frac {(A b-a B) (b d-a e)^2}{b^6 (a+b x)^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e)}{b^6 (a+b x)^2}+\frac {e (2 b B d+A b e-3 a B e)}{b^6 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(b d-a e) (b B d+2 A b e-3 a B e)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^2}{2 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^2 x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (2 b B d+A b e-3 a B e) (a+b x) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 151, normalized size = 0.81 \[ \frac {B \left (-5 a^3 e^2+2 a^2 b e (3 d-2 e x)+a b^2 \left (-d^2+8 d e x+4 e^2 x^2\right )+2 b^3 x \left (e^2 x^2-d^2\right )\right )+2 e (a+b x)^2 \log (a+b x) (-3 a B e+A b e+2 b B d)-A b (b d-a e) (3 a e+b (d+4 e x))}{2 b^4 (a+b x) \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 258, normalized size = 1.39 \[ \frac {2 \, B b^{3} e^{2} x^{3} + 4 \, B a b^{2} e^{2} x^{2} - {\left (B a b^{2} + A b^{3}\right )} d^{2} + 2 \, {\left (3 \, B a^{2} b - A a b^{2}\right )} d e - {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} - 2 \, {\left (B b^{3} d^{2} - 2 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d e + 2 \, {\left (B a^{2} b - A a b^{2}\right )} e^{2}\right )} x + 2 \, {\left (2 \, B a^{2} b d e - {\left (3 \, B a^{3} - A a^{2} b\right )} e^{2} + {\left (2 \, B b^{3} d e - {\left (3 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, B a b^{2} d e - {\left (3 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 303, normalized size = 1.63 \[ \frac {\left (2 A \,b^{3} e^{2} x^{2} \ln \left (b x +a \right )-6 B a \,b^{2} e^{2} x^{2} \ln \left (b x +a \right )+4 B \,b^{3} d e \,x^{2} \ln \left (b x +a \right )+2 B \,b^{3} e^{2} x^{3}+4 A a \,b^{2} e^{2} x \ln \left (b x +a \right )-12 B \,a^{2} b \,e^{2} x \ln \left (b x +a \right )+8 B a \,b^{2} d e x \ln \left (b x +a \right )+4 B a \,b^{2} e^{2} x^{2}+2 A \,a^{2} b \,e^{2} \ln \left (b x +a \right )+4 A a \,b^{2} e^{2} x -4 A \,b^{3} d e x -6 B \,a^{3} e^{2} \ln \left (b x +a \right )+4 B \,a^{2} b d e \ln \left (b x +a \right )-4 B \,a^{2} b \,e^{2} x +8 B a \,b^{2} d e x -2 B \,b^{3} d^{2} x +3 A \,a^{2} b \,e^{2}-2 A a \,b^{2} d e -A \,b^{3} d^{2}-5 B \,a^{3} e^{2}+6 B \,a^{2} b d e -B a \,b^{2} d^{2}\right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 277, normalized size = 1.49 \[ \frac {B e^{2} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {3 \, B a e^{2} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {2 \, B a^{2} e^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {6 \, B a^{2} e^{2} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (2 \, B d e + A e^{2}\right )} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {B d^{2} + 2 \, A d e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {A d^{2}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, B a^{3} e^{2}}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, {\left (2 \, B d e + A e^{2}\right )} a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, {\left (2 \, B d e + A e^{2}\right )} a^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (B d^{2} + 2 \, A d e\right )} a}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \left (d + e x\right )^{2}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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