Optimal. Leaf size=276 \[ \frac {b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (3 a e+b d)}{f^7 (m+7) \left (a+b x^2\right )}+\frac {3 a b \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+b d)}{f^5 (m+5) \left (a+b x^2\right )}+\frac {a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+3 b d)}{f^3 (m+3) \left (a+b x^2\right )}+\frac {b^3 e \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+9}}{f^9 (m+9) \left (a+b x^2\right )}+\frac {a^3 d \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left (a+b x^2\right )} \]
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Rubi [A] time = 0.15, 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 = {1250, 448} \[ \frac {a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+3 b d)}{f^3 (m+3) \left (a+b x^2\right )}+\frac {3 a b \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+b d)}{f^5 (m+5) \left (a+b x^2\right )}+\frac {b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (3 a e+b d)}{f^7 (m+7) \left (a+b x^2\right )}+\frac {a^3 d \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left (a+b x^2\right )}+\frac {b^3 e \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+9}}{f^9 (m+9) \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 448
Rule 1250
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 = 0.11, size = 112, normalized size = 0.41 \[ \frac {x \left (\left (a+b x^2\right )^2\right )^{3/2} (f x)^m \left (\frac {a^3 d}{m+1}+\frac {a^2 x^2 (a e+3 b d)}{m+3}+\frac {b^2 x^6 (3 a e+b d)}{m+7}+\frac {3 a b x^4 (a e+b d)}{m+5}+\frac {b^3 e x^8}{m+9}\right )}{\left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 381, normalized size = 1.38 \[ \frac {{\left ({\left (b^{3} e m^{4} + 16 \, b^{3} e m^{3} + 86 \, b^{3} e m^{2} + 176 \, b^{3} e m + 105 \, b^{3} e\right )} x^{9} + {\left ({\left (b^{3} d + 3 \, a b^{2} e\right )} m^{4} + 135 \, b^{3} d + 405 \, a b^{2} e + 18 \, {\left (b^{3} d + 3 \, a b^{2} e\right )} m^{3} + 104 \, {\left (b^{3} d + 3 \, a b^{2} e\right )} m^{2} + 222 \, {\left (b^{3} d + 3 \, a b^{2} e\right )} m\right )} x^{7} + 3 \, {\left ({\left (a b^{2} d + a^{2} b e\right )} m^{4} + 189 \, a b^{2} d + 189 \, a^{2} b e + 20 \, {\left (a b^{2} d + a^{2} b e\right )} m^{3} + 130 \, {\left (a b^{2} d + a^{2} b e\right )} m^{2} + 300 \, {\left (a b^{2} d + a^{2} b e\right )} m\right )} x^{5} + {\left ({\left (3 \, a^{2} b d + a^{3} e\right )} m^{4} + 945 \, a^{2} b d + 315 \, a^{3} e + 22 \, {\left (3 \, a^{2} b d + a^{3} e\right )} m^{3} + 164 \, {\left (3 \, a^{2} b d + a^{3} e\right )} m^{2} + 458 \, {\left (3 \, a^{2} b d + a^{3} e\right )} m\right )} x^{3} + {\left (a^{3} d m^{4} + 24 \, a^{3} d m^{3} + 206 \, a^{3} d m^{2} + 744 \, a^{3} d m + 945 \, a^{3} d\right )} x\right )} \left (f x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 1013, normalized size = 3.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 495, normalized size = 1.79 \[ \frac {\left (b^{3} e \,m^{4} x^{8}+16 b^{3} e \,m^{3} x^{8}+3 a \,b^{2} e \,m^{4} x^{6}+b^{3} d \,m^{4} x^{6}+86 b^{3} e \,m^{2} x^{8}+54 a \,b^{2} e \,m^{3} x^{6}+18 b^{3} d \,m^{3} x^{6}+176 b^{3} e m \,x^{8}+3 a^{2} b e \,m^{4} x^{4}+3 a \,b^{2} d \,m^{4} x^{4}+312 a \,b^{2} e \,m^{2} x^{6}+104 b^{3} d \,m^{2} x^{6}+105 b^{3} e \,x^{8}+60 a^{2} b e \,m^{3} x^{4}+60 a \,b^{2} d \,m^{3} x^{4}+666 a \,b^{2} e m \,x^{6}+222 b^{3} d m \,x^{6}+a^{3} e \,m^{4} x^{2}+3 a^{2} b d \,m^{4} x^{2}+390 a^{2} b e \,m^{2} x^{4}+390 a \,b^{2} d \,m^{2} x^{4}+405 a \,b^{2} e \,x^{6}+135 b^{3} d \,x^{6}+22 a^{3} e \,m^{3} x^{2}+66 a^{2} b d \,m^{3} x^{2}+900 a^{2} b e m \,x^{4}+900 a \,b^{2} d m \,x^{4}+a^{3} d \,m^{4}+164 a^{3} e \,m^{2} x^{2}+492 a^{2} b d \,m^{2} x^{2}+567 a^{2} b e \,x^{4}+567 a \,b^{2} d \,x^{4}+24 a^{3} d \,m^{3}+458 a^{3} e m \,x^{2}+1374 a^{2} b d m \,x^{2}+206 a^{3} d \,m^{2}+315 a^{3} e \,x^{2}+945 a^{2} b d \,x^{2}+744 a^{3} d m +945 a^{3} d \right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} x \left (f x \right )^{m}}{\left (m +9\right ) \left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right ) \left (b \,x^{2}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 243, normalized size = 0.88 \[ \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b^{3} f^{m} x^{7} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} a b^{2} f^{m} x^{5} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} a^{2} b f^{m} x^{3} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} a^{3} f^{m} x\right )} d x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} + \frac {{\left ({\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b^{3} f^{m} x^{9} + 3 \, {\left (m^{3} + 17 \, m^{2} + 87 \, m + 135\right )} a b^{2} f^{m} x^{7} + 3 \, {\left (m^{3} + 19 \, m^{2} + 111 \, m + 189\right )} a^{2} b f^{m} x^{5} + {\left (m^{3} + 21 \, m^{2} + 143 \, m + 315\right )} a^{3} f^{m} x^{3}\right )} e x^{m}}{m^{4} + 24 \, m^{3} + 206 \, m^{2} + 744 \, m + 945} \]
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 \[ \int {\left (f\,x\right )}^m\,\left (e\,x^2+d\right )\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \]
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 \[ \int \left (f x\right )^{m} \left (d + e x^{2}\right ) \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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