Optimal. Leaf size=138 \[ -\frac {2 (a+b x) (B d-A e)}{e \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}-\frac {2 (a+b x) (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {770, 78, 63, 208} \[ -\frac {2 (a+b x) (B d-A e)}{e \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}-\frac {2 (a+b x) (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^{3/2} \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)^{3/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 (B d-A e) (a+b x)}{e (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 (B d-A e) (a+b x)}{e (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 (A b-a B) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 (B d-A e) (a+b x)}{e (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (A b-a B) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 128, normalized size = 0.93 \[ \frac {2 (a+b x) \left (e \sqrt {d+e x} (a B-A b) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )-\sqrt {b} (b d-a e) (B d-A e)\right )}{\sqrt {b} e \sqrt {(a+b x)^2} \sqrt {d+e x} (b d-a e)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 363, normalized size = 2.63 \[ \left [\frac {{\left ({\left (B a - A b\right )} e^{2} x + {\left (B a - A b\right )} d e\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (B b^{2} d^{2} + A a b e^{2} - {\left (B a b + A b^{2}\right )} d e\right )} \sqrt {e x + d}}{b^{3} d^{3} e - 2 \, a b^{2} d^{2} e^{2} + a^{2} b d e^{3} + {\left (b^{3} d^{2} e^{2} - 2 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x}, -\frac {2 \, {\left ({\left ({\left (B a - A b\right )} e^{2} x + {\left (B a - A b\right )} d e\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (B b^{2} d^{2} + A a b e^{2} - {\left (B a b + A b^{2}\right )} d e\right )} \sqrt {e x + d}\right )}}{b^{3} d^{3} e - 2 \, a b^{2} d^{2} e^{2} + a^{2} b d e^{3} + {\left (b^{3} d^{2} e^{2} - 2 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 117, normalized size = 0.85 \[ -\frac {2 \, {\left (B a \mathrm {sgn}\left (b x + a\right ) - A b \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} {\left (b d - a e\right )}} - \frac {2 \, {\left (B d \mathrm {sgn}\left (b x + a\right ) - A e \mathrm {sgn}\left (b x + a\right )\right )}}{{\left (b d e - a e^{2}\right )} \sqrt {x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 148, normalized size = 1.07 \[ -\frac {2 \left (b x +a \right ) \left (\sqrt {e x +d}\, A b e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-\sqrt {e x +d}\, B a e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+\sqrt {\left (a e -b d \right ) b}\, A e -\sqrt {\left (a e -b d \right ) b}\, B d \right )}{\sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{\sqrt {{\left (b x + a\right )}^{2}} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {A+B\,x}{\sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {A + B x}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {\left (a + b x\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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