Optimal. Leaf size=160 \[ -\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (-a B e-A b e+2 b B d)}{e^3 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) \sqrt {d+e x}}+\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^3 (a+b x)} \]
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Rubi [A] time = 0.08, 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} \sqrt {d+e x} (-a B e-A b e+2 b B d)}{e^3 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) \sqrt {d+e x}}+\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 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)^{3/2}} \, 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)^{3/2}} \, 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)^{3/2}}+\frac {b (-2 b B d+A b e+a B e)}{e^2 \sqrt {d+e x}}+\frac {b^2 B \sqrt {d+e x}}{e^2}\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}}{e^3 (a+b x) \sqrt {d+e x}}-\frac {2 (2 b B d-A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac {2 b B (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 85, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (3 a e (-A e+2 B d+B e x)+3 A b e (2 d+e x)+b B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 20.63, size = 111, normalized size = 0.69 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-3 a A e^2+3 a B e (d+e x)+3 a B d e+3 A b e (d+e x)+3 A b d e-3 b B d^2-6 b B d (d+e x)+b B (d+e x)^2\right )}{3 e^2 \sqrt {d+e x} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 79, normalized size = 0.49 \begin {gather*} \frac {2 \, {\left (B b e^{2} x^{2} - 8 \, B b d^{2} - 3 \, A a e^{2} + 6 \, {\left (B a + A b\right )} d e - {\left (4 \, B b d e - 3 \, {\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 148, normalized size = 0.92 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b e^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, \sqrt {x e + d} B b d e^{6} \mathrm {sgn}\left (b x + a\right ) + 3 \, \sqrt {x e + d} B a e^{7} \mathrm {sgn}\left (b x + a\right ) + 3 \, \sqrt {x e + d} A b e^{7} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-9\right )} - \frac {2 \, {\left (B b d^{2} \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 )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 89, normalized size = 0.56 \begin {gather*} -\frac {2 \left (-B b \,x^{2} e^{2}-3 A b \,e^{2} x -3 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}-6 A b d e -6 B a d e +8 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \sqrt {e x +d}\, \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 = 0.77, size = 75, normalized size = 0.47 \begin {gather*} \frac {2 \, {\left (b e x + 2 \, b d - a e\right )} A}{\sqrt {e x + d} e^{2}} + \frac {2 \, {\left (b e^{2} x^{2} - 8 \, b d^{2} + 6 \, a d e - {\left (4 \, b d e - 3 \, a e^{2}\right )} x\right )} B}{3 \, \sqrt {e x + d} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 109, normalized size = 0.68 \begin {gather*} \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,B\,x^2}{3\,e}-\frac {6\,A\,a\,e^2+16\,B\,b\,d^2-12\,A\,b\,d\,e-12\,B\,a\,d\,e}{3\,b\,e^3}+\frac {x\,\left (6\,A\,b\,e^2+6\,B\,a\,e^2-8\,B\,b\,d\,e\right )}{3\,b\,e^3}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \end {gather*}
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 \begin {gather*} \int \frac {\left (A + B x\right ) \sqrt {\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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