Optimal. Leaf size=124 \[ -\frac {2 (d+e x)^{5/2} (-A c e-b B e+3 B c d)}{5 e^4}+\frac {2 (d+e x)^{3/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4}-\frac {2 d \sqrt {d+e x} (B d-A e) (c d-b e)}{e^4}+\frac {2 B c (d+e x)^{7/2}}{7 e^4} \]
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Rubi [A] time = 0.08, antiderivative size = 124, 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)^{5/2} (-A c e-b B e+3 B c d)}{5 e^4}+\frac {2 (d+e x)^{3/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4}-\frac {2 d \sqrt {d+e x} (B d-A e) (c d-b e)}{e^4}+\frac {2 B c (d+e x)^{7/2}}{7 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {d+e x}} \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e)}{e^3 \sqrt {d+e x}}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) \sqrt {d+e x}}{e^3}+\frac {(-3 B c d+b B e+A c e) (d+e x)^{3/2}}{e^3}+\frac {B c (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac {2 d (B d-A e) (c d-b e) \sqrt {d+e x}}{e^4}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{3/2}}{3 e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{5/2}}{5 e^4}+\frac {2 B c (d+e x)^{7/2}}{7 e^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 113, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {d+e x} \left (7 A e \left (5 b e (e x-2 d)+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+B \left (7 b e \left (8 d^2-4 d e x+3 e^2 x^2\right )-3 c \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )\right )}{105 e^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 141, normalized size = 1.14 \begin {gather*} \frac {2 \sqrt {d+e x} \left (35 A b e^2 (d+e x)-105 A b d e^2+105 A c d^2 e-70 A c d e (d+e x)+21 A c e (d+e x)^2+105 b B d^2 e-70 b B d e (d+e x)+21 b B e (d+e x)^2-105 B c d^3+105 B c d^2 (d+e x)-63 B c d (d+e x)^2+15 B c (d+e x)^3\right )}{105 e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 108, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (15 \, B c e^{3} x^{3} - 48 \, B c d^{3} - 70 \, A b d e^{2} + 56 \, {\left (B b + A c\right )} d^{2} e - 3 \, {\left (6 \, B c d e^{2} - 7 \, {\left (B b + A c\right )} e^{3}\right )} x^{2} + {\left (24 \, B c d^{2} e + 35 \, A b e^{3} - 28 \, {\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt {e x + d}}{105 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 167, normalized size = 1.35 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b 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 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 )} A c e^{\left (-2\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 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 = 0.05, size = 121, normalized size = 0.98 \begin {gather*} -\frac {2 \left (-15 B c \,x^{3} e^{3}-21 A c \,e^{3} x^{2}-21 B b \,e^{3} x^{2}+18 B c d \,e^{2} x^{2}-35 A b \,e^{3} x +28 A c d \,e^{2} x +28 B b d \,e^{2} x -24 B c \,d^{2} e x +70 A b d \,e^{2}-56 A c \,d^{2} e -56 B b \,d^{2} e +48 B c \,d^{3}\right ) \sqrt {e x +d}}{105 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 112, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} B c - 21 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\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 {3}{2}} - 105 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} \sqrt {e x + d}\right )}}{105 \, 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 = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{3\,e^4}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{5\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}}{e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 47.60, size = 430, normalized size = 3.47 \begin {gather*} \begin {cases} \frac {- \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 A c 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 A c \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}} - \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}} - \frac {2 B c d \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^{3}} - \frac {2 B c \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}}}{e} & \text {for}\: e \neq 0 \\\frac {\frac {A b x^{2}}{2} + \frac {B c x^{4}}{4} + \frac {x^{3} \left (A c + B b\right )}{3}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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