Optimal. Leaf size=149 \[ \frac {(2 c d-b e) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}+\frac {e \sqrt {b x+c x^2} \left (15 b^2 e^2+10 c e x (2 c d-b e)-54 b c d e+64 c^2 d^2\right )}{24 c^3}+\frac {e \sqrt {b x+c x^2} (d+e x)^2}{3 c} \]
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Rubi [A] time = 0.15, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {742, 779, 620, 206} \begin {gather*} \frac {e \sqrt {b x+c x^2} \left (15 b^2 e^2+10 c e x (2 c d-b e)-54 b c d e+64 c^2 d^2\right )}{24 c^3}+\frac {(2 c d-b e) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}+\frac {e \sqrt {b x+c x^2} (d+e x)^2}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 742
Rule 779
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\sqrt {b x+c x^2}} \, dx &=\frac {e (d+e x)^2 \sqrt {b x+c x^2}}{3 c}+\frac {\int \frac {(d+e x) \left (\frac {1}{2} d (6 c d-b e)+\frac {5}{2} e (2 c d-b e) x\right )}{\sqrt {b x+c x^2}} \, dx}{3 c}\\ &=\frac {e (d+e x)^2 \sqrt {b x+c x^2}}{3 c}+\frac {e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2+10 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{24 c^3}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c^3}\\ &=\frac {e (d+e x)^2 \sqrt {b x+c x^2}}{3 c}+\frac {e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2+10 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{24 c^3}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c^3}\\ &=\frac {e (d+e x)^2 \sqrt {b x+c x^2}}{3 c}+\frac {e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2+10 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{24 c^3}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 152, normalized size = 1.02 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} e \left (15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+\frac {3 \left (-5 b^3 e^3+18 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right ) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{24 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.03, size = 155, normalized size = 1.04 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (15 b^2 e^3-54 b c d e^2-10 b c e^3 x+72 c^2 d^2 e+36 c^2 d e^2 x+8 c^2 e^3 x^2\right )}{24 c^3}+\frac {\left (5 b^3 e^3-18 b^2 c d e^2+24 b c^2 d^2 e-16 c^3 d^3\right ) \log \left (-2 c^{7/2} \sqrt {b x+c x^2}+b c^3+2 c^4 x\right )}{16 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 296, normalized size = 1.99 \begin {gather*} \left [-\frac {3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\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^{3} x^{2} + 72 \, c^{3} d^{2} e - 54 \, b c^{2} d e^{2} + 15 \, b^{2} c e^{3} + 2 \, {\left (18 \, c^{3} d e^{2} - 5 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{4}}, -\frac {3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (8 \, c^{3} e^{3} x^{2} + 72 \, c^{3} d^{2} e - 54 \, b c^{2} d e^{2} + 15 \, b^{2} c e^{3} + 2 \, {\left (18 \, c^{3} d e^{2} - 5 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 147, normalized size = 0.99 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, x {\left (\frac {4 \, x e^{3}}{c} + \frac {18 \, c^{2} d e^{2} - 5 \, b c e^{3}}{c^{3}}\right )} + \frac {3 \, {\left (24 \, c^{2} d^{2} e - 18 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )}}{c^{3}}\right )} - \frac {{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 265, normalized size = 1.78 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x}\, e^{3} x^{2}}{3 c}-\frac {5 b^{3} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {7}{2}}}+\frac {9 b^{2} d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {5}{2}}}-\frac {3 b \,d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}+\frac {d^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}-\frac {5 \sqrt {c \,x^{2}+b x}\, b \,e^{3} x}{12 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x}\, d \,e^{2} x}{2 c}+\frac {5 \sqrt {c \,x^{2}+b x}\, b^{2} e^{3}}{8 c^{3}}-\frac {9 \sqrt {c \,x^{2}+b x}\, b d \,e^{2}}{4 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x}\, d^{2} e}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 260, normalized size = 1.74 \begin {gather*} \frac {\sqrt {c x^{2} + b x} e^{3} x^{2}}{3 \, c} + \frac {3 \, \sqrt {c x^{2} + b x} d e^{2} x}{2 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b e^{3} x}{12 \, c^{2}} + \frac {d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} - \frac {3 \, b d^{2} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} + \frac {9 \, b^{2} d e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {5 \, b^{3} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {7}{2}}} + \frac {3 \, \sqrt {c x^{2} + b x} d^{2} e}{c} - \frac {9 \, \sqrt {c x^{2} + b x} b d e^{2}}{4 \, c^{2}} + \frac {5 \, \sqrt {c x^{2} + b x} b^{2} e^{3}}{8 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\left (d + e x\right )^{3}}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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