Optimal. Leaf size=81 \[ -\frac {2 (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3}+\frac {2 \sqrt {d+e x} (b d-a e) (B d-A e)}{e^3}+\frac {2 b B (d+e x)^{5/2}}{5 e^3} \]
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Rubi [A] time = 0.03, antiderivative size = 81, 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} \[ -\frac {2 (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3}+\frac {2 \sqrt {d+e x} (b d-a e) (B d-A e)}{e^3}+\frac {2 b B (d+e x)^{5/2}}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{\sqrt {d+e x}} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e)}{e^2 \sqrt {d+e x}}+\frac {(-2 b B d+A b e+a B e) \sqrt {d+e x}}{e^2}+\frac {b B (d+e x)^{3/2}}{e^2}\right ) \, dx\\ &=\frac {2 (b d-a e) (B d-A e) \sqrt {d+e x}}{e^3}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{3/2}}{3 e^3}+\frac {2 b B (d+e x)^{5/2}}{5 e^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 68, normalized size = 0.84 \[ \frac {2 \sqrt {d+e x} \left (5 a e (3 A e-2 B d+B e x)+5 A b e (e x-2 d)+b B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 70, normalized size = 0.86 \[ \frac {2 \, {\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 15 \, A a e^{2} - 10 \, {\left (B a + A b\right )} d e - {\left (4 \, B b d e - 5 \, {\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 109, normalized size = 1.35 \[ \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a e^{\left (-1\right )} + 5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b e^{\left (-1\right )} + {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B b e^{\left (-2\right )} + 15 \, \sqrt {x e + d} A a\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 73, normalized size = 0.90 \[ \frac {2 \sqrt {e x +d}\, \left (3 B b \,x^{2} e^{2}+5 A b \,e^{2} x +5 B a \,e^{2} x -4 B b d e x +15 A a \,e^{2}-10 A b d e -10 B a d e +8 B b \,d^{2}\right )}{15 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 75, normalized size = 0.93 \[ \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B b - 5 \, {\left (2 \, B b d - {\left (B a + A b\right )} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} \sqrt {e x + d}\right )}}{15 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 80, normalized size = 0.99 \[ \frac {2\,\sqrt {d+e\,x}\,\left (3\,B\,b\,{\left (d+e\,x\right )}^2+15\,A\,a\,e^2+15\,B\,b\,d^2+5\,A\,b\,e\,\left (d+e\,x\right )+5\,B\,a\,e\,\left (d+e\,x\right )-10\,B\,b\,d\,\left (d+e\,x\right )-15\,A\,b\,d\,e-15\,B\,a\,d\,e\right )}{15\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.72, size = 311, normalized size = 3.84 \[ \begin {cases} \frac {- \frac {2 A a d}{\sqrt {d + e x}} - 2 A a \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {2 A b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 A b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {2 B a d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 B a \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {2 B b d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {2 B b \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {A a x + \frac {B b x^{3}}{3} + \frac {x^{2} \left (A b + B a\right )}{2}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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