Optimal. Leaf size=482 \[ \frac {\left (128 c^4 d^4+3 b^4 e^4-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+6 b^2 c e^3 (b d-2 a e)-2 c e \left (32 c^3 d^3-3 b^3 e^3-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3 e^5}+\frac {\left (16 c^2 d^2-6 b c d e-3 b^2 e^2-6 c e (2 c d+b e) x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{96 c^2 e^3}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c e}-\frac {\left (256 c^5 d^5+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+16 b c^2 e^3 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2} e^6}+\frac {d^2 \left (c d^2-b d e+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 e^6} \]
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Rubi [A]
time = 0.73, antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1265, 1667,
828, 857, 635, 212, 738} \begin {gather*} -\frac {\left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2} e^6}+\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{96 c^2 e^3}+\frac {\sqrt {a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{256 c^3 e^5}+\frac {d^2 \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e^6}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 828
Rule 857
Rule 1265
Rule 1667
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx,x,x^2\right )\\ &=\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c e}+\frac {\text {Subst}\left (\int \frac {\left (-\frac {5}{2} b d e-\frac {5}{2} e (2 c d+b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx,x,x^2\right )}{10 c e^2}\\ &=\frac {\left (16 c^2 d^2-6 b c d e-3 b^2 e^2-6 c e (2 c d+b e) x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{96 c^2 e^3}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c e}-\frac {\text {Subst}\left (\int \frac {\left (-\frac {5}{4} d e \left (6 b^2 c d e+8 a c^2 d e+3 b^3 e^2-4 b c \left (4 c d^2+3 a e^2\right )\right )+\frac {5}{4} e \left (32 c^3 d^3-3 b^3 e^3-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx,x,x^2\right )}{80 c^2 e^4}\\ &=\frac {\left (128 c^4 d^4+3 b^4 e^4-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+6 b^2 c e^3 (b d-2 a e)-2 c e \left (32 c^3 d^3-3 b^3 e^3-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3 e^5}+\frac {\left (16 c^2 d^2-6 b c d e-3 b^2 e^2-6 c e (2 c d+b e) x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{96 c^2 e^3}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c e}+\frac {\text {Subst}\left (\int \frac {-\frac {5}{8} d e \left (6 b^4 c d e^3+3 b^5 e^4+8 b^3 c e^2 \left (2 c d^2-3 a e^2\right )-16 b^2 c^2 d e \left (10 c d^2+3 a e^2\right )-32 a c^3 d e \left (4 c d^2+5 a e^2\right )+16 b c^2 \left (8 c^2 d^4+20 a c d^2 e^2+3 a^2 e^4\right )\right )-\frac {5}{8} e \left (256 c^5 d^5+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+16 b c^2 e^3 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{320 c^3 e^6}\\ &=\frac {\left (128 c^4 d^4+3 b^4 e^4-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+6 b^2 c e^3 (b d-2 a e)-2 c e \left (32 c^3 d^3-3 b^3 e^3-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3 e^5}+\frac {\left (16 c^2 d^2-6 b c d e-3 b^2 e^2-6 c e (2 c d+b e) x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{96 c^2 e^3}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c e}+\frac {\left (d^2 \left (c d^2-b d e+a e^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 e^6}-\frac {\left (256 c^5 d^5+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+16 b c^2 e^3 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{512 c^3 e^6}\\ &=\frac {\left (128 c^4 d^4+3 b^4 e^4-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+6 b^2 c e^3 (b d-2 a e)-2 c e \left (32 c^3 d^3-3 b^3 e^3-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3 e^5}+\frac {\left (16 c^2 d^2-6 b c d e-3 b^2 e^2-6 c e (2 c d+b e) x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{96 c^2 e^3}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c e}-\frac {\left (d^2 \left (c d^2-b d e+a e^2\right )^2\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^2}{\sqrt {a+b x^2+c x^4}}\right )}{e^6}-\frac {\left (256 c^5 d^5+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+16 b c^2 e^3 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )}{256 c^3 e^6}\\ &=\frac {\left (128 c^4 d^4+3 b^4 e^4-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+6 b^2 c e^3 (b d-2 a e)-2 c e \left (32 c^3 d^3-3 b^3 e^3-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3 e^5}+\frac {\left (16 c^2 d^2-6 b c d e-3 b^2 e^2-6 c e (2 c d+b e) x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{96 c^2 e^3}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c e}-\frac {\left (256 c^5 d^5+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+16 b c^2 e^3 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2} e^6}+\frac {d^2 \left (c d^2-b d e+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 e^6}\\ \end {align*}
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Mathematica [A]
time = 10.66, size = 545, normalized size = 1.13 \begin {gather*} \frac {1280 d^2 \left (a+b x^2+c x^4\right )^{3/2}-\frac {480 d e \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{c}+\frac {768 e^2 \left (a+b x^2+c x^4\right )^{5/2}}{c}-\frac {90 \left (b^2-4 a c\right ) d e \left (-2 \sqrt {c} \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}+\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )\right )}{c^{5/2}}+\frac {15 b e^2 \left (-16 \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}+3 \left (b^2-4 a c\right ) \left (\frac {2 \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{c}+\frac {\left (-b^2+4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{c^{3/2}}\right )\right )}{c^2}-\frac {240 d^2 \left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )+2 \sqrt {c} \left (e \sqrt {a+b x^2+c x^4} \left (-b^2 e^2+4 c^2 d \left (-2 d+e x^2\right )-2 c e \left (-5 b d+4 a e+b e x^2\right )\right )+8 c \left (c d^2+e (-b d+a e)\right )^{3/2} \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x^2+b e x^2}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+b x^2+c x^4}}\right )\right )\right )}{c^{3/2} e^3}}{7680 e^3} \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. \(1984\) vs.
\(2(450)=900\).
time = 0.17, size = 1985, normalized size = 4.12
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1798\) |
default | \(\text {Expression too large to display}\) | \(1985\) |
elliptic | \(\text {Expression too large to display}\) | \(2068\) |
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 [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{e\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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