Optimal. Leaf size=212 \[ \frac {(a+b x) (A b-a B)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^2}-\frac {(a+b x) (B d-A e)}{2 e \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}+\frac {b (a+b x) (A b-a B) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {b (a+b x) (A b-a B) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
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Rubi [A] time = 0.15, antiderivative size = 212, 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} \[ \frac {(a+b x) (A b-a B)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^2}-\frac {(a+b x) (B d-A e)}{2 e \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}+\frac {b (a+b x) (A b-a B) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {b (a+b x) (A b-a B) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {A+B x}{\left (a b+b^2 x\right ) (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 \left (\frac {b (A b-a B)}{(b d-a e)^3 (a+b x)}+\frac {B d-A e}{b (b d-a e) (d+e x)^3}+\frac {(-A b+a B) e}{b (b d-a e)^2 (d+e x)^2}+\frac {(-A b+a B) e}{(b d-a e)^3 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(B d-A e) (a+b x)}{2 e (b d-a e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x)}{(b d-a e)^2 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (A b-a B) (a+b x) \log (a+b x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (A b-a B) (a+b x) \log (d+e x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 131, normalized size = 0.62 \[ \frac {(a+b x) \left (\frac {2 (A b-a B)}{(d+e x) (b d-a e)^2}+\frac {B d-A e}{e (d+e x)^2 (a e-b d)}+\frac {2 b (A b-a B) \log (a+b x)}{(b d-a e)^3}-\frac {2 b (A b-a B) \log (d+e x)}{(b d-a e)^3}\right )}{2 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 343, normalized size = 1.62 \[ -\frac {B b^{2} d^{3} - 3 \, A b^{2} d^{2} e - A a^{2} e^{3} - {\left (B a^{2} - 4 \, A a b\right )} d e^{2} + 2 \, {\left ({\left (B a b - A b^{2}\right )} d e^{2} - {\left (B a^{2} - A a b\right )} e^{3}\right )} x + 2 \, {\left ({\left (B a b - A b^{2}\right )} e^{3} x^{2} + 2 \, {\left (B a b - A b^{2}\right )} d e^{2} x + {\left (B a b - A b^{2}\right )} d^{2} e\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B a b - A b^{2}\right )} e^{3} x^{2} + 2 \, {\left (B a b - A b^{2}\right )} d e^{2} x + {\left (B a b - A b^{2}\right )} d^{2} e\right )} \log \left (e x + d\right )}{2 \, {\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4} + {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{2} + 2 \, {\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 306, normalized size = 1.44 \[ -\frac {{\left (B a b^{2} \mathrm {sgn}\left (b x + a\right ) - A b^{3} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} + \frac {{\left (B a b e \mathrm {sgn}\left (b x + a\right ) - A b^{2} e \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} - \frac {{\left (B b^{2} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, A b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) - B a^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, A a b d e^{2} \mathrm {sgn}\left (b x + a\right ) - A a^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (B a b d e^{2} \mathrm {sgn}\left (b x + a\right ) - A b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - B a^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + A a b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-1\right )}}{2 \, {\left (b d - a e\right )}^{3} {\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 321, normalized size = 1.51 \[ -\frac {\left (b x +a \right ) \left (2 A \,b^{2} e^{3} x^{2} \ln \left (b x +a \right )-2 A \,b^{2} e^{3} x^{2} \ln \left (e x +d \right )-2 B a b \,e^{3} x^{2} \ln \left (b x +a \right )+2 B a b \,e^{3} x^{2} \ln \left (e x +d \right )+4 A \,b^{2} d \,e^{2} x \ln \left (b x +a \right )-4 A \,b^{2} d \,e^{2} x \ln \left (e x +d \right )-4 B a b d \,e^{2} x \ln \left (b x +a \right )+4 B a b d \,e^{2} x \ln \left (e x +d \right )-2 A a b \,e^{3} x +2 A \,b^{2} d^{2} e \ln \left (b x +a \right )-2 A \,b^{2} d^{2} e \ln \left (e x +d \right )+2 A \,b^{2} d \,e^{2} x +2 B \,a^{2} e^{3} x -2 B a b \,d^{2} e \ln \left (b x +a \right )+2 B a b \,d^{2} e \ln \left (e x +d \right )-2 B a b d \,e^{2} x +A \,a^{2} e^{3}-4 A a b d \,e^{2}+3 A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}-B \,b^{2} d^{3}\right )}{2 \sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right )^{3} \left (e x +d \right )^{2} e} \]
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 \[ \int \frac {A+B\,x}{\sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.99, size = 558, normalized size = 2.63 \[ - \frac {b \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{2} e - A b^{3} d + B a^{2} b e + B a b^{2} d - \frac {a^{4} b e^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} + \frac {4 a^{3} b^{2} d e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} - \frac {6 a^{2} b^{3} d^{2} e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} + \frac {4 a b^{4} d^{3} e \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} - \frac {b^{5} d^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}}}{- 2 A b^{3} e + 2 B a b^{2} e} \right )}}{\left (a e - b d\right )^{3}} + \frac {b \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{2} e - A b^{3} d + B a^{2} b e + B a b^{2} d + \frac {a^{4} b e^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} - \frac {4 a^{3} b^{2} d e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} + \frac {6 a^{2} b^{3} d^{2} e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} - \frac {4 a b^{4} d^{3} e \left (- A b + B a\right )}{\left (a e - b d\right )^{3}} + \frac {b^{5} d^{4} \left (- A b + B a\right )}{\left (a e - b d\right )^{3}}}{- 2 A b^{3} e + 2 B a b^{2} e} \right )}}{\left (a e - b d\right )^{3}} + \frac {- A a e^{2} + 3 A b d e - B a d e - B b d^{2} + x \left (2 A b e^{2} - 2 B a e^{2}\right )}{2 a^{2} d^{2} e^{3} - 4 a b d^{3} e^{2} + 2 b^{2} d^{4} e + x^{2} \left (2 a^{2} e^{5} - 4 a b d e^{4} + 2 b^{2} d^{2} e^{3}\right ) + x \left (4 a^{2} d e^{4} - 8 a b d^{2} e^{3} + 4 b^{2} d^{3} e^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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