Optimal. Leaf size=198 \[ -\frac {\sqrt {\frac {d x^4}{c}+1} (e x)^{m+1} \left (2 b^2 c^2 (m+1)-(m+3) \left (2 a^2 d^2-(m+1) (b c-a d)^2\right )\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{4};\frac {m+5}{4};-\frac {d x^4}{c}\right )}{2 c d^2 e (m+1) (m+3) \sqrt {c+d x^4}}+\frac {(e x)^{m+1} (b c-a d)^2}{2 c d^2 e \sqrt {c+d x^4}}+\frac {b^2 \sqrt {c+d x^4} (e x)^{m+1}}{d^2 e (m+3)} \]
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Rubi [A] time = 0.18, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {463, 459, 365, 364} \[ -\frac {\sqrt {\frac {d x^4}{c}+1} (e x)^{m+1} \left (2 b^2 c^2 (m+1)-(m+3) \left (2 a^2 d^2-(m+1) (b c-a d)^2\right )\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{4};\frac {m+5}{4};-\frac {d x^4}{c}\right )}{2 c d^2 e (m+1) (m+3) \sqrt {c+d x^4}}+\frac {(e x)^{m+1} (b c-a d)^2}{2 c d^2 e \sqrt {c+d x^4}}+\frac {b^2 \sqrt {c+d x^4} (e x)^{m+1}}{d^2 e (m+3)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 459
Rule 463
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (a+b x^4\right )^2}{\left (c+d x^4\right )^{3/2}} \, dx &=\frac {(b c-a d)^2 (e x)^{1+m}}{2 c d^2 e \sqrt {c+d x^4}}-\frac {\int \frac {(e x)^m \left (-2 a^2 d^2+(b c-a d)^2 (1+m)-2 b^2 c d x^4\right )}{\sqrt {c+d x^4}} \, dx}{2 c d^2}\\ &=\frac {(b c-a d)^2 (e x)^{1+m}}{2 c d^2 e \sqrt {c+d x^4}}+\frac {b^2 (e x)^{1+m} \sqrt {c+d x^4}}{d^2 e (3+m)}-\frac {\left (-a^2 d^2 (1-m)-2 a b c d (1+m)+\frac {b^2 c^2 (1+m) (5+m)}{3+m}\right ) \int \frac {(e x)^m}{\sqrt {c+d x^4}} \, dx}{2 c d^2}\\ &=\frac {(b c-a d)^2 (e x)^{1+m}}{2 c d^2 e \sqrt {c+d x^4}}+\frac {b^2 (e x)^{1+m} \sqrt {c+d x^4}}{d^2 e (3+m)}-\frac {\left (\left (-a^2 d^2 (1-m)-2 a b c d (1+m)+\frac {b^2 c^2 (1+m) (5+m)}{3+m}\right ) \sqrt {1+\frac {d x^4}{c}}\right ) \int \frac {(e x)^m}{\sqrt {1+\frac {d x^4}{c}}} \, dx}{2 c d^2 \sqrt {c+d x^4}}\\ &=\frac {(b c-a d)^2 (e x)^{1+m}}{2 c d^2 e \sqrt {c+d x^4}}+\frac {b^2 (e x)^{1+m} \sqrt {c+d x^4}}{d^2 e (3+m)}+\frac {\left (a^2 d^2 (1-m)+2 a b c d (1+m)-\frac {b^2 c^2 (1+m) (5+m)}{3+m}\right ) (e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} \, _2F_1\left (\frac {1}{2},\frac {1+m}{4};\frac {5+m}{4};-\frac {d x^4}{c}\right )}{2 c d^2 e (1+m) \sqrt {c+d x^4}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 167, normalized size = 0.84 \[ \frac {x \sqrt {\frac {d x^4}{c}+1} (e x)^m \left (a^2 \left (m^2+14 m+45\right ) \, _2F_1\left (\frac {3}{2},\frac {m+1}{4};\frac {m+5}{4};-\frac {d x^4}{c}\right )+b (m+1) x^4 \left (2 a (m+9) \, _2F_1\left (\frac {3}{2},\frac {m+5}{4};\frac {m+9}{4};-\frac {d x^4}{c}\right )+b (m+5) x^4 \, _2F_1\left (\frac {3}{2},\frac {m+9}{4};\frac {m+13}{4};-\frac {d x^4}{c}\right )\right )\right )}{c (m+1) (m+5) (m+9) \sqrt {c+d x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} \sqrt {d x^{4} + c} \left (e x\right )^{m}}{d^{2} x^{8} + 2 \, c d x^{4} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{2} \left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{2} \left (e x \right )^{m}}{\left (d \,x^{4}+c \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{2} \left (e x\right )^{m}}{{\left (d x^{4} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x\right )}^m\,{\left (b\,x^4+a\right )}^2}{{\left (d\,x^4+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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