Optimal. Leaf size=126 \[ -\frac {2 (d+e x)^{7/2} (-A c e-b B e+3 B c d)}{7 e^4}+\frac {2 (d+e x)^{5/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac {2 d (d+e x)^{3/2} (B d-A e) (c d-b e)}{3 e^4}+\frac {2 B c (d+e x)^{9/2}}{9 e^4} \]
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Rubi [A] time = 0.08, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{7/2} (-A c e-b B e+3 B c d)}{7 e^4}+\frac {2 (d+e x)^{5/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac {2 d (d+e x)^{3/2} (B d-A e) (c d-b e)}{3 e^4}+\frac {2 B c (d+e x)^{9/2}}{9 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right ) \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e) \sqrt {d+e x}}{e^3}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{3/2}}{e^3}+\frac {(-3 B c d+b B e+A c e) (d+e x)^{5/2}}{e^3}+\frac {B c (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac {2 d (B d-A e) (c d-b e) (d+e x)^{3/2}}{3 e^4}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{5/2}}{5 e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{7/2}}{7 e^4}+\frac {2 B c (d+e x)^{9/2}}{9 e^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 113, normalized size = 0.90 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (3 A e \left (7 b e (3 e x-2 d)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+B \left (3 b e \left (8 d^2-12 d e x+15 e^2 x^2\right )+c \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )\right )}{315 e^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 141, normalized size = 1.12 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (63 A b e^2 (d+e x)-105 A b d e^2+105 A c d^2 e-126 A c d e (d+e x)+45 A c e (d+e x)^2+105 b B d^2 e-126 b B d e (d+e x)+45 b B e (d+e x)^2-105 B c d^3+189 B c d^2 (d+e x)-135 B c d (d+e x)^2+35 B c (d+e x)^3\right )}{315 e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 148, normalized size = 1.17 \begin {gather*} \frac {2 \, {\left (35 \, B c e^{4} x^{4} - 16 \, B c d^{4} - 42 \, A b d^{2} e^{2} + 24 \, {\left (B b + A c\right )} d^{3} e + 5 \, {\left (B c d e^{3} + 9 \, {\left (B b + A c\right )} e^{4}\right )} x^{3} - 3 \, {\left (2 \, B c d^{2} e^{2} - 21 \, A b e^{4} - 3 \, {\left (B b + A c\right )} d e^{3}\right )} x^{2} + {\left (8 \, B c d^{3} e + 21 \, A b d e^{3} - 12 \, {\left (B b + A c\right )} d^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 386, normalized size = 3.06 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b d 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 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 c d e^{\left (-2\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 )} B c d e^{\left (-3\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 b 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 )} B b e^{\left (-2\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 c e^{\left (-2\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 c e^{\left (-3\right )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 121, normalized size = 0.96 \begin {gather*} -\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (-35 B c \,x^{3} e^{3}-45 A c \,e^{3} x^{2}-45 B b \,e^{3} x^{2}+30 B c d \,e^{2} x^{2}-63 A b \,e^{3} x +36 A c d \,e^{2} x +36 B b d \,e^{2} x -24 B c \,d^{2} e x +42 A b d \,e^{2}-24 A c \,d^{2} e -24 B b \,d^{2} e +16 B c \,d^{3}\right )}{315 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 112, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B c - 45 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{315 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 111, normalized size = 0.88 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{5\,e^4}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{7\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.18, size = 146, normalized size = 1.16 \begin {gather*} \frac {2 \left (\frac {B c \left (d + e x\right )^{\frac {9}{2}}}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A c e + B b e - 3 B c d\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}\right )}{5 e^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- A b d e^{2} + A c d^{2} e + B b d^{2} e - B c d^{3}\right )}{3 e^{3}}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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