Optimal. Leaf size=276 \[ \frac {a^3 d (f x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f (1+m) \left (a+b x^2\right )}+\frac {a^2 (3 b d+a e) (f x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^3 (3+m) \left (a+b x^2\right )}+\frac {3 a b (b d+a e) (f x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^5 (5+m) \left (a+b x^2\right )}+\frac {b^2 (b d+3 a e) (f x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^7 (7+m) \left (a+b x^2\right )}+\frac {b^3 e (f x)^{9+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^9 (9+m) \left (a+b x^2\right )} \]
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Rubi [A]
time = 0.10, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1264, 459}
\begin {gather*} \frac {b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (3 a e+b d)}{f^7 (m+7) \left (a+b x^2\right )}+\frac {3 a b \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+b d)}{f^5 (m+5) \left (a+b x^2\right )}+\frac {a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+3 b d)}{f^3 (m+3) \left (a+b x^2\right )}+\frac {b^3 e \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+9}}{f^9 (m+9) \left (a+b x^2\right )}+\frac {a^3 d \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rule 1264
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (f x)^m \left (a b+b^2 x^2\right )^3 \left (d+e x^2\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^3 b^3 d (f x)^m+\frac {a^2 b^3 (3 b d+a e) (f x)^{2+m}}{f^2}+\frac {3 a b^4 (b d+a e) (f x)^{4+m}}{f^4}+\frac {b^5 (b d+3 a e) (f x)^{6+m}}{f^6}+\frac {b^6 e (f x)^{8+m}}{f^8}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {a^3 d (f x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f (1+m) \left (a+b x^2\right )}+\frac {a^2 (3 b d+a e) (f x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^3 (3+m) \left (a+b x^2\right )}+\frac {3 a b (b d+a e) (f x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^5 (5+m) \left (a+b x^2\right )}+\frac {b^2 (b d+3 a e) (f x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^7 (7+m) \left (a+b x^2\right )}+\frac {b^3 e (f x)^{9+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^9 (9+m) \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 112, normalized size = 0.41 \begin {gather*} \frac {x (f x)^m \left (\left (a+b x^2\right )^2\right )^{3/2} \left (\frac {a^3 d}{1+m}+\frac {a^2 (3 b d+a e) x^2}{3+m}+\frac {3 a b (b d+a e) x^4}{5+m}+\frac {b^2 (b d+3 a e) x^6}{7+m}+\frac {b^3 e x^8}{9+m}\right )}{\left (a+b x^2\right )^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. \(494\) vs.
\(2(221)=442\).
time = 0.02, size = 495, normalized size = 1.79
method | result | size |
gosper | \(\frac {x \left (b^{3} e \,m^{4} x^{8}+16 b^{3} e \,m^{3} x^{8}+3 a \,b^{2} e \,m^{4} x^{6}+b^{3} d \,m^{4} x^{6}+86 b^{3} e \,m^{2} x^{8}+54 a \,b^{2} e \,m^{3} x^{6}+18 b^{3} d \,m^{3} x^{6}+176 m \,x^{8} b^{3} e +3 a^{2} b e \,m^{4} x^{4}+3 a \,b^{2} d \,m^{4} x^{4}+312 a \,b^{2} e \,m^{2} x^{6}+104 b^{3} d \,m^{2} x^{6}+105 b^{3} e \,x^{8}+60 a^{2} b e \,m^{3} x^{4}+60 a \,b^{2} d \,m^{3} x^{4}+666 a \,b^{2} e \,x^{6} m +222 b^{3} d \,x^{6} m +a^{3} e \,m^{4} x^{2}+3 a^{2} b d \,m^{4} x^{2}+390 a^{2} b e \,m^{2} x^{4}+390 a \,b^{2} d \,m^{2} x^{4}+405 a \,b^{2} e \,x^{6}+135 b^{3} d \,x^{6}+22 a^{3} e \,m^{3} x^{2}+66 a^{2} b d \,m^{3} x^{2}+900 a^{2} b e \,x^{4} m +900 a \,b^{2} d \,x^{4} m +a^{3} d \,m^{4}+164 a^{3} e \,m^{2} x^{2}+492 a^{2} b d \,m^{2} x^{2}+567 a^{2} b e \,x^{4}+567 a \,b^{2} d \,x^{4}+24 a^{3} d \,m^{3}+458 a^{3} e \,x^{2} m +1374 a^{2} b d \,x^{2} m +206 a^{3} d \,m^{2}+315 a^{3} e \,x^{2}+945 a^{2} b d \,x^{2}+744 a^{3} d m +945 a^{3} d \right ) \left (f x \right )^{m} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right ) \left (b \,x^{2}+a \right )^{3}}\) | \(495\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (b^{3} e \,m^{4} x^{8}+16 b^{3} e \,m^{3} x^{8}+3 a \,b^{2} e \,m^{4} x^{6}+b^{3} d \,m^{4} x^{6}+86 b^{3} e \,m^{2} x^{8}+54 a \,b^{2} e \,m^{3} x^{6}+18 b^{3} d \,m^{3} x^{6}+176 m \,x^{8} b^{3} e +3 a^{2} b e \,m^{4} x^{4}+3 a \,b^{2} d \,m^{4} x^{4}+312 a \,b^{2} e \,m^{2} x^{6}+104 b^{3} d \,m^{2} x^{6}+105 b^{3} e \,x^{8}+60 a^{2} b e \,m^{3} x^{4}+60 a \,b^{2} d \,m^{3} x^{4}+666 a \,b^{2} e \,x^{6} m +222 b^{3} d \,x^{6} m +a^{3} e \,m^{4} x^{2}+3 a^{2} b d \,m^{4} x^{2}+390 a^{2} b e \,m^{2} x^{4}+390 a \,b^{2} d \,m^{2} x^{4}+405 a \,b^{2} e \,x^{6}+135 b^{3} d \,x^{6}+22 a^{3} e \,m^{3} x^{2}+66 a^{2} b d \,m^{3} x^{2}+900 a^{2} b e \,x^{4} m +900 a \,b^{2} d \,x^{4} m +a^{3} d \,m^{4}+164 a^{3} e \,m^{2} x^{2}+492 a^{2} b d \,m^{2} x^{2}+567 a^{2} b e \,x^{4}+567 a \,b^{2} d \,x^{4}+24 a^{3} d \,m^{3}+458 a^{3} e \,x^{2} m +1374 a^{2} b d \,x^{2} m +206 a^{3} d \,m^{2}+315 a^{3} e \,x^{2}+945 a^{2} b d \,x^{2}+744 a^{3} d m +945 a^{3} d \right ) x \left (f x \right )^{m}}{\left (b \,x^{2}+a \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(495\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 246, normalized size = 0.89 \begin {gather*} \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b^{3} f^{m} x^{7} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} a b^{2} f^{m} x^{5} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} a^{2} b f^{m} x^{3} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} a^{3} f^{m} x\right )} d x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} + \frac {{\left ({\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b^{3} f^{m} x^{9} + 3 \, {\left (m^{3} + 17 \, m^{2} + 87 \, m + 135\right )} a b^{2} f^{m} x^{7} + 3 \, {\left (m^{3} + 19 \, m^{2} + 111 \, m + 189\right )} a^{2} b f^{m} x^{5} + {\left (m^{3} + 21 \, m^{2} + 143 \, m + 315\right )} a^{3} f^{m} x^{3}\right )} e^{\left (m \log \left (x\right ) + 1\right )}}{m^{4} + 24 \, m^{3} + 206 \, m^{2} + 744 \, m + 945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 390, normalized size = 1.41 \begin {gather*} \frac {{\left ({\left (b^{3} d m^{4} + 18 \, b^{3} d m^{3} + 104 \, b^{3} d m^{2} + 222 \, b^{3} d m + 135 \, b^{3} d\right )} x^{7} + 3 \, {\left (a b^{2} d m^{4} + 20 \, a b^{2} d m^{3} + 130 \, a b^{2} d m^{2} + 300 \, a b^{2} d m + 189 \, a b^{2} d\right )} x^{5} + 3 \, {\left (a^{2} b d m^{4} + 22 \, a^{2} b d m^{3} + 164 \, a^{2} b d m^{2} + 458 \, a^{2} b d m + 315 \, a^{2} b d\right )} x^{3} + {\left (a^{3} d m^{4} + 24 \, a^{3} d m^{3} + 206 \, a^{3} d m^{2} + 744 \, a^{3} d m + 945 \, a^{3} d\right )} x + {\left ({\left (b^{3} m^{4} + 16 \, b^{3} m^{3} + 86 \, b^{3} m^{2} + 176 \, b^{3} m + 105 \, b^{3}\right )} x^{9} + 3 \, {\left (a b^{2} m^{4} + 18 \, a b^{2} m^{3} + 104 \, a b^{2} m^{2} + 222 \, a b^{2} m + 135 \, a b^{2}\right )} x^{7} + 3 \, {\left (a^{2} b m^{4} + 20 \, a^{2} b m^{3} + 130 \, a^{2} b m^{2} + 300 \, a^{2} b m + 189 \, a^{2} b\right )} x^{5} + {\left (a^{3} m^{4} + 22 \, a^{3} m^{3} + 164 \, a^{3} m^{2} + 458 \, a^{3} m + 315 \, a^{3}\right )} x^{3}\right )} e\right )} \left (f x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \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 \left (f x\right )^{m} \left (d + e x^{2}\right ) \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \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. 1013 vs.
\(2 (225) = 450\).
time = 5.42, size = 1013, normalized size = 3.67 \begin {gather*} \frac {\left (f x\right )^{m} b^{3} m^{4} x^{9} e \mathrm {sgn}\left (b x^{2} + a\right ) + 16 \, \left (f x\right )^{m} b^{3} m^{3} x^{9} e \mathrm {sgn}\left (b x^{2} + a\right ) + \left (f x\right )^{m} b^{3} d m^{4} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, \left (f x\right )^{m} a b^{2} m^{4} x^{7} e \mathrm {sgn}\left (b x^{2} + a\right ) + 86 \, \left (f x\right )^{m} b^{3} m^{2} x^{9} e \mathrm {sgn}\left (b x^{2} + a\right ) + 18 \, \left (f x\right )^{m} b^{3} d m^{3} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 54 \, \left (f x\right )^{m} a b^{2} m^{3} x^{7} e \mathrm {sgn}\left (b x^{2} + a\right ) + 176 \, \left (f x\right )^{m} b^{3} m x^{9} e \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, \left (f x\right )^{m} a b^{2} d m^{4} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 104 \, \left (f x\right )^{m} b^{3} d m^{2} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, \left (f x\right )^{m} a^{2} b m^{4} x^{5} e \mathrm {sgn}\left (b x^{2} + a\right ) + 312 \, \left (f x\right )^{m} a b^{2} m^{2} x^{7} e \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, \left (f x\right )^{m} b^{3} x^{9} e \mathrm {sgn}\left (b x^{2} + a\right ) + 60 \, \left (f x\right )^{m} a b^{2} d m^{3} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 222 \, \left (f x\right )^{m} b^{3} d m x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 60 \, \left (f x\right )^{m} a^{2} b m^{3} x^{5} e \mathrm {sgn}\left (b x^{2} + a\right ) + 666 \, \left (f x\right )^{m} a b^{2} m x^{7} e \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, \left (f x\right )^{m} a^{2} b d m^{4} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 390 \, \left (f x\right )^{m} a b^{2} d m^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 135 \, \left (f x\right )^{m} b^{3} d x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + \left (f x\right )^{m} a^{3} m^{4} x^{3} e \mathrm {sgn}\left (b x^{2} + a\right ) + 390 \, \left (f x\right )^{m} a^{2} b m^{2} x^{5} e \mathrm {sgn}\left (b x^{2} + a\right ) + 405 \, \left (f x\right )^{m} a b^{2} x^{7} e \mathrm {sgn}\left (b x^{2} + a\right ) + 66 \, \left (f x\right )^{m} a^{2} b d m^{3} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 900 \, \left (f x\right )^{m} a b^{2} d m x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 22 \, \left (f x\right )^{m} a^{3} m^{3} x^{3} e \mathrm {sgn}\left (b x^{2} + a\right ) + 900 \, \left (f x\right )^{m} a^{2} b m x^{5} e \mathrm {sgn}\left (b x^{2} + a\right ) + \left (f x\right )^{m} a^{3} d m^{4} x \mathrm {sgn}\left (b x^{2} + a\right ) + 492 \, \left (f x\right )^{m} a^{2} b d m^{2} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 567 \, \left (f x\right )^{m} a b^{2} d x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 164 \, \left (f x\right )^{m} a^{3} m^{2} x^{3} e \mathrm {sgn}\left (b x^{2} + a\right ) + 567 \, \left (f x\right )^{m} a^{2} b x^{5} e \mathrm {sgn}\left (b x^{2} + a\right ) + 24 \, \left (f x\right )^{m} a^{3} d m^{3} x \mathrm {sgn}\left (b x^{2} + a\right ) + 1374 \, \left (f x\right )^{m} a^{2} b d m x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 458 \, \left (f x\right )^{m} a^{3} m x^{3} e \mathrm {sgn}\left (b x^{2} + a\right ) + 206 \, \left (f x\right )^{m} a^{3} d m^{2} x \mathrm {sgn}\left (b x^{2} + a\right ) + 945 \, \left (f x\right )^{m} a^{2} b d x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 315 \, \left (f x\right )^{m} a^{3} x^{3} e \mathrm {sgn}\left (b x^{2} + a\right ) + 744 \, \left (f x\right )^{m} a^{3} d m x \mathrm {sgn}\left (b x^{2} + a\right ) + 945 \, \left (f x\right )^{m} a^{3} d x \mathrm {sgn}\left (b x^{2} + a\right )}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \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 {\left (f\,x\right )}^m\,\left (e\,x^2+d\right )\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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