Optimal. Leaf size=99 \[ -\frac {b^2 (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}}+\frac {(b+2 c x) \sqrt {b x+c x^2} (2 c d-b e)}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c} \]
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Rubi [A] time = 0.04, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {640, 612, 620, 206} \[ -\frac {b^2 (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}}+\frac {(b+2 c x) \sqrt {b x+c x^2} (2 c d-b e)}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rubi steps
\begin {align*} \int (d+e x) \sqrt {b x+c x^2} \, dx &=\frac {e \left (b x+c x^2\right )^{3/2}}{3 c}+\frac {(2 c d-b e) \int \sqrt {b x+c x^2} \, dx}{2 c}\\ &=\frac {(2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac {\left (b^2 (2 c d-b e)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c^2}\\ &=\frac {(2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac {\left (b^2 (2 c d-b e)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c^2}\\ &=\frac {(2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac {b^2 (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 108, normalized size = 1.09 \[ \frac {\sqrt {x (b+c x)} \left (\frac {3 b^{3/2} (b e-2 c d) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} \left (-3 b^2 e+2 b c (3 d+e x)+4 c^2 x (3 d+2 e x)\right )\right )}{24 c^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 205, normalized size = 2.07 \[ \left [-\frac {3 \, {\left (2 \, b^{2} c d - b^{3} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (8 \, c^{3} e x^{2} + 6 \, b c^{2} d - 3 \, b^{2} c e + 2 \, {\left (6 \, c^{3} d + b c^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{3}}, \frac {3 \, {\left (2 \, b^{2} c d - b^{3} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (8 \, c^{3} e x^{2} + 6 \, b c^{2} d - 3 \, b^{2} c e + 2 \, {\left (6 \, c^{3} d + b c^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 108, normalized size = 1.09 \[ \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, x e + \frac {6 \, c^{2} d + b c e}{c^{2}}\right )} x + \frac {3 \, {\left (2 \, b c d - b^{2} e\right )}}{c^{2}}\right )} + \frac {{\left (2 \, b^{2} c d - b^{3} e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 157, normalized size = 1.59 \[ \frac {b^{3} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {5}{2}}}-\frac {b^{2} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{2}+b x}\, b e x}{4 c}+\frac {\sqrt {c \,x^{2}+b x}\, d x}{2}-\frac {\sqrt {c \,x^{2}+b x}\, b^{2} e}{8 c^{2}}+\frac {\sqrt {c \,x^{2}+b x}\, b d}{4 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} e}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 154, normalized size = 1.56 \[ \frac {1}{2} \, \sqrt {c x^{2} + b x} d x - \frac {\sqrt {c x^{2} + b x} b e x}{4 \, c} - \frac {b^{2} d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} + \frac {b^{3} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {5}{2}}} + \frac {\sqrt {c x^{2} + b x} b d}{4 \, c} - \frac {\sqrt {c x^{2} + b x} b^{2} e}{8 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} e}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 127, normalized size = 1.28 \[ d\,\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )-\frac {b^2\,d\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}}+\frac {b^3\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {e\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x \left (b + c x\right )} \left (d + e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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