Optimal. Leaf size=216 \[ \frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{8 c e^3}-\frac {(2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e^4}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e} \]
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Rubi [A] time = 0.24, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {734, 814, 843, 620, 206, 724} \begin {gather*} \frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{8 c e^3}-\frac {(2 c d-b e) \left (-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^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e^4}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 724
Rule 734
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx &=\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\int \frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{d+e x} \, dx}{2 e}\\ &=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}+\frac {\int \frac {-\frac {1}{2} b d \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-\frac {1}{2} (2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 c e^3}\\ &=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}+\frac {\left (d^2 (c d-b e)^2\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{e^4}-\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c e^4}\\ &=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\left (2 d^2 (c d-b e)^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{e^4}-\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e-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 e^4}\\ &=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 225, normalized size = 1.04 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} e \sqrt {x} \left (3 b^2 e^2+2 b c e (7 e x-15 d)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-\frac {3 \left (b^3 e^3+6 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 {\frac {c x}{b}+1}}+\frac {48 c^{3/2} d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {b+c x}}\right )}{24 c^{3/2} e^4 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.27, size = 238, normalized size = 1.10 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (3 b^2 e^2-30 b c d e+14 b c e^2 x+24 c^2 d^2-12 c^2 d e x+8 c^2 e^2 x^2\right )}{24 c e^3}+\frac {\left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right ) \log \left (-2 c^{3/2} \sqrt {b x+c x^2}+b c+2 c^2 x\right )}{16 c^{3/2} e^4}+\frac {2 \sqrt {c d-b e} \left (c d^{5/2}-b d^{3/2} e\right ) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 901, normalized size = 4.17 \begin {gather*} \left [\frac {3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 48 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + 2 \, {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{2} e^{4}}, \frac {96 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + 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} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{2} e^{4}}, \frac {3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - 24 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{2} e^{4}}, \frac {48 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + 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} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{2} e^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1090, normalized size = 5.05
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{d+e\,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 (x \left (b + c x\right )\right )^{\frac {3}{2}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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