Optimal. Leaf size=79 \[ -\frac {2 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{e^3}-\frac {2 (b d-a e) (B d-A e)}{e^3 \sqrt {d+e x}}+\frac {2 b B (d+e x)^{3/2}}{3 e^3} \]
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Rubi [A] time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {2 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{e^3}-\frac {2 (b d-a e) (B d-A e)}{e^3 \sqrt {d+e x}}+\frac {2 b B (d+e x)^{3/2}}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^{3/2}}+\frac {-2 b B d+A b e+a B e}{e^2 \sqrt {d+e x}}+\frac {b B \sqrt {d+e x}}{e^2}\right ) \, dx\\ &=-\frac {2 (b d-a e) (B d-A e)}{e^3 \sqrt {d+e x}}-\frac {2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{e^3}+\frac {2 b B (d+e x)^{3/2}}{3 e^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 68, normalized size = 0.86 \[ \frac {6 a e (-A e+2 B d+B e x)+6 A b e (2 d+e x)+2 b B \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 79, normalized size = 1.00 \[ \frac {2 \, {\left (B b e^{2} x^{2} - 8 \, B b d^{2} - 3 \, A a e^{2} + 6 \, {\left (B a + A b\right )} d e - {\left (4 \, B b d e - 3 \, {\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 100, normalized size = 1.27 \[ \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b e^{6} - 6 \, \sqrt {x e + d} B b d e^{6} + 3 \, \sqrt {x e + d} B a e^{7} + 3 \, \sqrt {x e + d} A b e^{7}\right )} e^{\left (-9\right )} - \frac {2 \, {\left (B b d^{2} - B a d e - A b d e + A a e^{2}\right )} e^{\left (-3\right )}}{\sqrt {x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 73, normalized size = 0.92 \[ -\frac {2 \left (-B b \,x^{2} e^{2}-3 A b \,e^{2} x -3 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}-6 A b d e -6 B a d e +8 B b \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 82, normalized size = 1.04 \[ \frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} B b - 3 \, {\left (2 \, B b d - {\left (B a + A b\right )} e\right )} \sqrt {e x + d}}{e^{2}} - \frac {3 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )}}{\sqrt {e x + d} e^{2}}\right )}}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 79, normalized size = 1.00 \[ \frac {\frac {2\,B\,b\,{\left (d+e\,x\right )}^2}{3}-2\,A\,a\,e^2-2\,B\,b\,d^2+2\,A\,b\,e\,\left (d+e\,x\right )+2\,B\,a\,e\,\left (d+e\,x\right )-4\,B\,b\,d\,\left (d+e\,x\right )+2\,A\,b\,d\,e+2\,B\,a\,d\,e}{e^3\,\sqrt {d+e\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.01, size = 76, normalized size = 0.96 \[ \frac {2 B b \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} + \frac {\sqrt {d + e x} \left (2 A b e + 2 B a e - 4 B b d\right )}{e^{3}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )}{e^{3} \sqrt {d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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