Optimal. Leaf size=510 \[ -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2048 \left (c d^2-b d e+a e^2\right )^{11/2}} \]
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Rubi [A]
time = 0.52, antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {758, 848, 820,
734, 738, 212} \begin {gather*} -\frac {e \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (4 a e+17 b d)+21 b^2 e^2+68 c^2 d^2\right )}{280 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}+\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}-\frac {3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{1024 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 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 )}{2048 \left (a e^2-b d e+c d^2\right )^{11/2}}-\frac {3 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{28 (d+e x)^6 \left (a e^2-b d e+c d^2\right )^2}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 (d+e x)^7 \left (a e^2-b d e+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 734
Rule 738
Rule 758
Rule 820
Rule 848
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx &=-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {\int \frac {\left (\frac {1}{2} (-14 c d+9 b e)+2 c e x\right ) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx}{7 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}+\frac {\int \frac {\left (\frac {3}{4} \left (56 c^2 d^2+21 b^2 e^2-2 c e (31 b d+8 a e)\right )-\frac {9}{2} c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{42 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{16 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}-\frac {\left (3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx}{256 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2048 \left (c d^2-b d e+a e^2\right )^5}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}-\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \text {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 )}{1024 \left (c d^2-b d e+a e^2\right )^5}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{1024 \left (c d^2-b d e+a e^2\right )^5 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{7 \left (c d^2-b d e+a e^2\right ) (d+e x)^7}-\frac {3 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{28 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}-\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) \left (a+b x+c x^2\right )^{5/2}}{280 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\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 )}{2048 \left (c d^2-b d e+a e^2\right )^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 15.22, size = 430, normalized size = 0.84 \begin {gather*} -\frac {\frac {e (a+x (b+c x))^{5/2}}{(d+e x)^7}+\frac {3 e (2 c d-b e) (a+x (b+c x))^{5/2}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^6}+\frac {e \left (68 c^2 d^2+21 b^2 e^2-4 c e (17 b d+4 a e)\right ) (a+x (b+c x))^{5/2}}{40 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)^5}+\frac {7 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \left (\frac {2 (-b d+2 a e-2 c d x+b e x) (a+x (b+c x))^{3/2}}{(d+e x)^4}+3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )\right )}{256 \left (c d^2+e (-b d+a e)\right )^3}}{7 \left (c d^2+e (-b d+a e)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(15719\) vs.
\(2(480)=960\).
time = 0.82, size = 15720, normalized size = 30.82
method | result | size |
default | \(\text {Expression too large to display}\) | \(15720\) |
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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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 22291 vs.
\(2 (502) = 1004\).
time = 8.92, size = 22291, normalized size = 43.71 \begin {gather*} \text {Too large to display} \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+a\right )}^{3/2}}{{\left (d+e\,x\right )}^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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