Optimal. Leaf size=589 \[ \frac {d^3 (f x)^{m+1} \left (a+b \sec ^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \sec ^{-1}(c x)\right )}{f^7 (m+7)}-\frac {b e^3 x \sqrt {c^2 x^2-1} (f x)^{m+5}}{c f^5 (m+6) (m+7) \sqrt {c^2 x^2}}-\frac {b e^2 x \sqrt {c^2 x^2-1} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^3 f^3 (m+4) (m+5) (m+6) (m+7) \sqrt {c^2 x^2}}-\frac {b e x \sqrt {c^2 x^2-1} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^5 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \sqrt {c^2 x^2}}-\frac {b x \sqrt {1-c^2 x^2} (f x)^{m+1} \left (\frac {c^6 d^3 (m+2) (m+4) (m+6)}{m+1}+\frac {e (m+1) \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{(m+3) (m+5) (m+7)}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{c^5 f (m+1) (m+2) (m+4) (m+6) \sqrt {c^2 x^2} \sqrt {c^2 x^2-1}} \]
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Rubi [A] time = 2.42, antiderivative size = 570, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {270, 5238, 1809, 1267, 459, 365, 364} \[ \frac {3 d^2 e (f x)^{m+3} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} \left (a+b \sec ^{-1}(c x)\right )}{f (m+1)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \sec ^{-1}(c x)\right )}{f^7 (m+7)}-\frac {b c x \sqrt {1-c^2 x^2} (f x)^{m+1} \left (\frac {e \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^6 (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+\frac {d^3}{(m+1)^2}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{f \sqrt {c^2 x^2} \sqrt {c^2 x^2-1}}-\frac {b e x \sqrt {c^2 x^2-1} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^5 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \sqrt {c^2 x^2}}-\frac {b e^2 x \sqrt {c^2 x^2-1} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^3 f^3 (m+4) (m+5) (m+6) (m+7) \sqrt {c^2 x^2}}-\frac {b e^3 x \sqrt {c^2 x^2-1} (f x)^{m+5}}{c f^5 (m+6) (m+7) \sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 364
Rule 365
Rule 459
Rule 1267
Rule 1809
Rule 5238
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac {d^3 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sec ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {(b c x) \int \frac {(f x)^m \left (\frac {d^3}{1+m}+\frac {3 d^2 e x^2}{3+m}+\frac {3 d e^2 x^4}{5+m}+\frac {e^3 x^6}{7+m}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sec ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {(b x) \int \frac {(f x)^m \left (\frac {c^2 d^3 (6+m)}{1+m}+\frac {3 c^2 d^2 e (6+m) x^2}{3+m}+\frac {e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x^4}{(5+m) (7+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c (6+m) \sqrt {c^2 x^2}}\\ &=-\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}-\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sec ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {(b x) \int \frac {(f x)^m \left (\frac {c^4 d^3 (4+m) (6+m)}{1+m}+\frac {e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c^3 (4+m) (6+m) \sqrt {c^2 x^2}}\\ &=-\frac {b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}-\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}-\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sec ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b \left (\frac {c^4 d^3 (4+m) (6+m)}{1+m}+\frac {e (1+m) \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{c^2 (2+m) (3+m) (5+m) (7+m)}\right ) x\right ) \int \frac {(f x)^m}{\sqrt {-1+c^2 x^2}} \, dx}{c^3 (4+m) (6+m) \sqrt {c^2 x^2}}\\ &=-\frac {b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}-\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}-\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sec ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b \left (\frac {c^4 d^3 (4+m) (6+m)}{1+m}+\frac {e (1+m) \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{c^2 (2+m) (3+m) (5+m) (7+m)}\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^m}{\sqrt {1-c^2 x^2}} \, dx}{c^3 (4+m) (6+m) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}\\ &=-\frac {b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}-\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}-\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \sec ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sec ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sec ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sec ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {b \left (\frac {c^6 d^3}{(1+m)^2}+\frac {e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{(2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}\right ) x (f x)^{1+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{c^5 f \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}\\ \end {align*}
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Mathematica [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \sec ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} + {\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \operatorname {arcsec}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]
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 \[ \int {\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 9.23, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{3} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )\, dx \]
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 \[ \frac {a e^{3} f^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a d e^{2} f^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a d^{2} e f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d^{3}}{f {\left (m + 1\right )}} + \frac {{\left ({\left (b e^{3} f^{m} m^{3} + 9 \, b e^{3} f^{m} m^{2} + 23 \, b e^{3} f^{m} m + 15 \, b e^{3} f^{m}\right )} x^{7} + 3 \, {\left (b d e^{2} f^{m} m^{3} + 11 \, b d e^{2} f^{m} m^{2} + 31 \, b d e^{2} f^{m} m + 21 \, b d e^{2} f^{m}\right )} x^{5} + 3 \, {\left (b d^{2} e f^{m} m^{3} + 13 \, b d^{2} e f^{m} m^{2} + 47 \, b d^{2} e f^{m} m + 35 \, b d^{2} e f^{m}\right )} x^{3} + {\left (b d^{3} f^{m} m^{3} + 15 \, b d^{3} f^{m} m^{2} + 71 \, b d^{3} f^{m} m + 105 \, b d^{3} f^{m}\right )} x\right )} x^{m} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} \int \frac {{\left (b d^{3} f^{m} m^{3} + 15 \, b d^{3} f^{m} m^{2} + {\left (b e^{3} f^{m} m^{3} + 9 \, b e^{3} f^{m} m^{2} + 23 \, b e^{3} f^{m} m + 15 \, b e^{3} f^{m}\right )} x^{6} + 71 \, b d^{3} f^{m} m + 105 \, b d^{3} f^{m} + 3 \, {\left (b d e^{2} f^{m} m^{3} + 11 \, b d e^{2} f^{m} m^{2} + 31 \, b d e^{2} f^{m} m + 21 \, b d e^{2} f^{m}\right )} x^{4} + 3 \, {\left (b d^{2} e f^{m} m^{3} + 13 \, b d^{2} e f^{m} m^{2} + 47 \, b d^{2} e f^{m} m + 35 \, b d^{2} e f^{m}\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1} x^{m}}{m^{4} + 16 \, m^{3} - {\left (c^{2} m^{4} + 16 \, c^{2} m^{3} + 86 \, c^{2} m^{2} + 176 \, c^{2} m + 105 \, c^{2}\right )} x^{2} + 86 \, m^{2} + 176 \, m + 105}\,{d x}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]
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 \[ \int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^3\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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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