Optimal. Leaf size=303 \[ -\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {c} e^5}+\frac {\sqrt {a e^2-b d e+c d^2} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e^5}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^4}+\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)} \]
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Rubi [A] time = 0.48, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {812, 814, 843, 621, 206, 724} \[ \frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^4}-\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {c} e^5}+\frac {\sqrt {a e^2-b d e+c d^2} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e^5}+\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx &=\frac {(8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {\int \frac {\left (8 b c d-3 b^2 e-4 a c e+8 c (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{2 e^2}\\ &=\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{2 e^4}+\frac {(8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}+\frac {\int \frac {2 c \left (e (b d-2 a e) \left (8 b c d-3 b^2 e-4 a c e\right )-2 d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )-2 c (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 c e^4}\\ &=\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{2 e^4}+\frac {(8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {\left ((2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{4 e^5}+\frac {\left (2 c d (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )+2 c e \left (e (b d-2 a e) \left (8 b c d-3 b^2 e-4 a c e\right )-2 d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 c e^5}\\ &=\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{2 e^4}+\frac {(8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {\left ((2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{2 e^5}-\frac {\left (2 c d (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )+2 c e \left (e (b d-2 a e) \left (8 b c d-3 b^2 e-4 a c e\right )-2 d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{4 c e^5}\\ &=\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{2 e^4}+\frac {(8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {(2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {c} e^5}+\frac {\sqrt {c d^2-b d e+a e^2} \left (16 c^2 d^2-16 b c d e+3 b^2 e^2+4 a c e^2\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 e^5}\\ \end {align*}
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Mathematica [A] time = 1.06, size = 395, normalized size = 1.30 \[ \frac {\frac {(a+x (b+c x))^{3/2} \left (c e (-2 a e+11 b d-3 b e x)-3 b^2 e^2+c^2 \left (6 d e x-8 d^2\right )\right )}{3 e^2}+\frac {-2 c^2 e \sqrt {a+x (b+c x)} \left (e (a e-b d)+c d^2\right ) \left (4 c e (a e-5 b d+b e x)+5 b^2 e^2+8 c^2 d (2 d-e x)\right )+2 c^2 \left (4 c e (a e-4 b d)+3 b^2 e^2+16 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )+c^{3/2} (2 c d-b e) \left (e (a e-b d)+c d^2\right ) \left (4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{4 c^2 e^5}+\frac {(a+x (b+c x))^{5/2} (b e-2 c d)}{d+e x}}{e (b d-a e)-c d^2} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 6898, normalized size = 22.77 \[ \text {output too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\left (b+2\,c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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