Optimal. Leaf size=83 \[ -\frac {2 (d+e x)^{7/2} (-a B e-A b e+2 b B d)}{7 e^3}+\frac {2 (d+e x)^{5/2} (b d-a e) (B d-A e)}{5 e^3}+\frac {2 b B (d+e x)^{9/2}}{9 e^3} \]
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Rubi [A] time = 0.03, 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} \[ -\frac {2 (d+e x)^{7/2} (-a B e-A b e+2 b B d)}{7 e^3}+\frac {2 (d+e x)^{5/2} (b d-a e) (B d-A e)}{5 e^3}+\frac {2 b B (d+e x)^{9/2}}{9 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x) (A+B x) (d+e x)^{3/2} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e) (d+e x)^{3/2}}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^{5/2}}{e^2}+\frac {b B (d+e x)^{7/2}}{e^2}\right ) \, dx\\ &=\frac {2 (b d-a e) (B d-A e) (d+e x)^{5/2}}{5 e^3}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{7/2}}{7 e^3}+\frac {2 b B (d+e x)^{9/2}}{9 e^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 70, normalized size = 0.84 \[ \frac {2 (d+e x)^{5/2} \left (9 a e (7 A e-2 B d+5 B e x)+9 A b e (5 e x-2 d)+b B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 149, normalized size = 1.80 \[ \frac {2 \, {\left (35 \, B b e^{4} x^{4} + 8 \, B b d^{4} + 63 \, A a d^{2} e^{2} - 18 \, {\left (B a + A b\right )} d^{3} e + 5 \, {\left (10 \, B b d e^{3} + 9 \, {\left (B a + A b\right )} e^{4}\right )} x^{3} + 3 \, {\left (B b d^{2} e^{2} + 21 \, A a e^{4} + 24 \, {\left (B a + A b\right )} d e^{3}\right )} x^{2} - {\left (4 \, B b d^{3} e - 126 \, A a d e^{3} - 9 \, {\left (B a + A b\right )} d^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.20, size = 499, normalized size = 6.01 \[ \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a d^{2} e^{\left (-1\right )} + 105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b d^{2} e^{\left (-1\right )} + 21 \, {\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^{2} e^{\left (-2\right )} + 42 \, {\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 d e^{\left (-1\right )} + 42 \, {\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 d e^{\left (-1\right )} + 18 \, {\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 d e^{\left (-2\right )} + 315 \, \sqrt {x e + d} A a d^{2} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a d + 9 \, {\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 a e^{\left (-1\right )} + 9 \, {\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 )} A b e^{\left (-1\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B b e^{\left (-2\right )} + 21 \, {\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 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 = 0.88 \[ \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (35 B b \,x^{2} e^{2}+45 A b \,e^{2} x +45 B a \,e^{2} x -20 B b d e x +63 A a \,e^{2}-18 A b d e -18 B a d e +8 B b \,d^{2}\right )}{315 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 75, normalized size = 0.90 \[ \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b - 45 \, {\left (2 \, B b d - {\left (B a + A b\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{315 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 80, normalized size = 0.96 \[ \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (35\,B\,b\,{\left (d+e\,x\right )}^2+63\,A\,a\,e^2+63\,B\,b\,d^2+45\,A\,b\,e\,\left (d+e\,x\right )+45\,B\,a\,e\,\left (d+e\,x\right )-90\,B\,b\,d\,\left (d+e\,x\right )-63\,A\,b\,d\,e-63\,B\,a\,d\,e\right )}{315\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.75, size = 318, normalized size = 3.83 \[ A a d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 A a \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {2 A b d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {2 A b \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {2 B a d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {2 B a \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {2 B b d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 B b \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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