Optimal. Leaf size=72 \[ \frac {2 \sqrt {a+b x} \left (a^2 e-a b d+b^2 c\right )}{b^3}+\frac {2 (a+b x)^{3/2} (b d-2 a e)}{3 b^3}+\frac {2 e (a+b x)^{5/2}}{5 b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 72, 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 = {698} \[ \frac {2 \sqrt {a+b x} \left (a^2 e-a b d+b^2 c\right )}{b^3}+\frac {2 (a+b x)^{3/2} (b d-2 a e)}{3 b^3}+\frac {2 e (a+b x)^{5/2}}{5 b^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2}{\sqrt {a+b x}} \, dx &=\int \left (\frac {b^2 c-a b d+a^2 e}{b^2 \sqrt {a+b x}}+\frac {(b d-2 a e) \sqrt {a+b x}}{b^2}+\frac {e (a+b x)^{3/2}}{b^2}\right ) \, dx\\ &=\frac {2 \left (b^2 c-a b d+a^2 e\right ) \sqrt {a+b x}}{b^3}+\frac {2 (b d-2 a e) (a+b x)^{3/2}}{3 b^3}+\frac {2 e (a+b x)^{5/2}}{5 b^3}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 53, normalized size = 0.74 \[ \frac {2 \sqrt {a+b x} \left (8 a^2 e-2 a b (5 d+2 e x)+b^2 (15 c+x (5 d+3 e x))\right )}{15 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 53, normalized size = 0.74 \[ \frac {2 \, {\left (3 \, b^{2} e x^{2} + 15 \, b^{2} c - 10 \, a b d + 8 \, a^{2} e + {\left (5 \, b^{2} d - 4 \, a b e\right )} x\right )} \sqrt {b x + a}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 78, normalized size = 1.08 \[ \frac {2 \, {\left (15 \, \sqrt {b x + a} c + \frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {{\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}}\right )}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 53, normalized size = 0.74 \[ \frac {2 \sqrt {b x +a}\, \left (3 e \,x^{2} b^{2}-4 a b e x +5 b^{2} d x +8 a^{2} e -10 a b d +15 b^{2} c \right )}{15 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 77, normalized size = 1.07 \[ \frac {2 \, {\left (15 \, \sqrt {b x + a} c + \frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {{\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}}\right )}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 58, normalized size = 0.81 \[ \frac {2\,\sqrt {a+b\,x}\,\left (3\,e\,{\left (a+b\,x\right )}^2+15\,b^2\,c+15\,a^2\,e-10\,a\,e\,\left (a+b\,x\right )+5\,b\,d\,\left (a+b\,x\right )-15\,a\,b\,d\right )}{15\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.05, size = 223, normalized size = 3.10 \[ \begin {cases} \frac {- \frac {2 a c}{\sqrt {a + b x}} - \frac {2 a d \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right )}{b} - \frac {2 a e \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b^{2}} - 2 c \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right ) - \frac {2 d \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b} - \frac {2 e \left (- \frac {a^{3}}{\sqrt {a + b x}} - 3 a^{2} \sqrt {a + b x} + a \left (a + b x\right )^{\frac {3}{2}} - \frac {\left (a + b x\right )^{\frac {5}{2}}}{5}\right )}{b^{2}}}{b} & \text {for}\: b \neq 0 \\\frac {c x + \frac {d x^{2}}{2} + \frac {e x^{3}}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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