Optimal. Leaf size=160 \[ \frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{e^3 (a+b x) \sqrt {d+e x}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{3 e^3 (a+b x) (d+e x)^{3/2}}+\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^3 (a+b x)} \]
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Rubi [A] time = 0.09, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} \frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{e^3 (a+b x) \sqrt {d+e x}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{3 e^3 (a+b x) (d+e x)^{3/2}}+\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^3 (a+b x)} \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)^{5/2}} \, 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)^{5/2}} \, 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)^{5/2}}+\frac {b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^{3/2}}+\frac {b^2 B}{e^2 \sqrt {d+e x}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e) (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^{3/2}}+\frac {2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}}+\frac {2 b B \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 86, normalized size = 0.54 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (a e (A e+2 B d+3 B e x)+A b e (2 d+3 e x)-b B \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )}{3 e^3 (a+b x) (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 26.32, size = 110, normalized size = 0.69 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-a A e^2-3 a B e (d+e x)+a B d e-3 A b e (d+e x)+A b d e-b B d^2+6 b B d (d+e x)+3 b B (d+e x)^2\right )}{3 e^2 (d+e x)^{3/2} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 91, normalized size = 0.57 \begin {gather*} \frac {2 \, {\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} - A a e^{2} - 2 \, {\left (B a + A b\right )} d e + 3 \, {\left (4 \, B b d e - {\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 136, normalized size = 0.85 \begin {gather*} 2 \, \sqrt {x e + d} B b e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {2 \, {\left (6 \, {\left (x e + d\right )} B b d \mathrm {sgn}\left (b x + a\right ) - B b d^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (x e + d\right )} B a e \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (x e + d\right )} A b e \mathrm {sgn}\left (b x + a\right ) + B a d e \mathrm {sgn}\left (b x + a\right ) + A b d e \mathrm {sgn}\left (b x + a\right ) - A a e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 0.55 \begin {gather*} -\frac {2 \left (-3 B b \,x^{2} e^{2}+3 A b \,e^{2} x +3 B a \,e^{2} x -12 B b d e x +A a \,e^{2}+2 A b d e +2 B a d e -8 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right ) e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 96, normalized size = 0.60 \begin {gather*} -\frac {2 \, {\left (3 \, b e x + 2 \, b d + a e\right )} A}{3 \, {\left (e^{3} x + d e^{2}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (3 \, b e^{2} x^{2} + 8 \, b d^{2} - 2 \, a d e + 3 \, {\left (4 \, b d e - a e^{2}\right )} x\right )} B}{3 \, {\left (e^{4} x + d e^{3}\right )} \sqrt {e x + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 146, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,A\,a\,e^2-16\,B\,b\,d^2+4\,A\,b\,d\,e+4\,B\,a\,d\,e}{3\,b\,e^4}-\frac {2\,B\,x^2}{e^2}+\frac {x\,\left (6\,A\,b\,e^2+6\,B\,a\,e^2-24\,B\,b\,d\,e\right )}{3\,b\,e^4}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (3\,a\,e^4+3\,b\,d\,e^3\right )\,\sqrt {d+e\,x}}{3\,b\,e^4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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