Optimal. Leaf size=83 \[ -\frac {2 (d+e x)^{5/2} (-a B e-A b e+2 b B d)}{5 e^3}+\frac {2 (d+e x)^{3/2} (b d-a e) (B d-A e)}{3 e^3}+\frac {2 b B (d+e x)^{7/2}}{7 e^3} \]
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Rubi [A] time = 0.04, antiderivative size = 83, 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} \[ -\frac {2 (d+e x)^{5/2} (-a B e-A b e+2 b B d)}{5 e^3}+\frac {2 (d+e x)^{3/2} (b d-a e) (B d-A e)}{3 e^3}+\frac {2 b B (d+e x)^{7/2}}{7 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x) (A+B x) \sqrt {d+e x} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e) \sqrt {d+e x}}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^2}+\frac {b B (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac {2 (b d-a e) (B d-A e) (d+e x)^{3/2}}{3 e^3}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{5/2}}{5 e^3}+\frac {2 b B (d+e x)^{7/2}}{7 e^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 70, normalized size = 0.84 \[ \frac {2 (d+e x)^{3/2} \left (7 a e (5 A e-2 B d+3 B e x)+7 A b e (3 e x-2 d)+b B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 108, normalized size = 1.30 \[ \frac {2 \, {\left (15 \, B b e^{3} x^{3} + 8 \, B b d^{3} + 35 \, A a d e^{2} - 14 \, {\left (B a + A b\right )} d^{2} e + 3 \, {\left (B b d e^{2} + 7 \, {\left (B a + A b\right )} e^{3}\right )} x^{2} - {\left (4 \, B b d^{2} e - 35 \, A a e^{3} - 7 \, {\left (B a + A b\right )} d e^{2}\right )} x\right )} \sqrt {e x + d}}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.21, size = 274, normalized size = 3.30 \[ \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a d e^{\left (-1\right )} + 35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b d e^{\left (-1\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B b d e^{\left (-2\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a e^{\left (-1\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A b e^{\left (-1\right )} + 3 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B b e^{\left (-2\right )} + 105 \, \sqrt {x e + d} A a d + 35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 73, normalized size = 0.88 \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (15 B b \,x^{2} e^{2}+21 A b \,e^{2} x +21 B a \,e^{2} x -12 B b d e x +35 A a \,e^{2}-14 A b d e -14 B a d e +8 B b \,d^{2}\right )}{105 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 75, normalized size = 0.90 \[ \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} B b - 21 \, {\left (2 \, B b d - {\left (B a + A b\right )} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 80, normalized size = 0.96 \[ \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (15\,B\,b\,{\left (d+e\,x\right )}^2+35\,A\,a\,e^2+35\,B\,b\,d^2+21\,A\,b\,e\,\left (d+e\,x\right )+21\,B\,a\,e\,\left (d+e\,x\right )-42\,B\,b\,d\,\left (d+e\,x\right )-35\,A\,b\,d\,e-35\,B\,a\,d\,e\right )}{105\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.70, size = 94, normalized size = 1.13 \[ \frac {2 \left (\frac {B b \left (d + e x\right )^{\frac {7}{2}}}{7 e^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A b e + B a e - 2 B b d\right )}{5 e^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a e^{2} - A b d e - B a d e + B b d^{2}\right )}{3 e^{2}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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