Optimal. Leaf size=484 \[ \frac {d^3 (f x)^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (m+7)}+\frac {b e^3 \sqrt {1-c^2 x^2} (f x)^{m+6}}{c f^6 (m+7)^2}+\frac {b e^2 \sqrt {1-c^2 x^2} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^3 f^4 (m+5)^2 (m+7)^2}+\frac {b e \sqrt {1-c^2 x^2} (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^5 f^2 (m+3)^2 (m+5)^2 (m+7)^2}-\frac {b (f x)^{m+2} \left (\frac {c^6 d^3 (m+3) (m+5) (m+7)}{m+1}+\frac {e (m+2) \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{(m+3) (m+5) (m+7)}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{c^5 f^2 (m+2) (m+3) (m+5) (m+7)} \]
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Rubi [A] time = 2.38, antiderivative size = 455, normalized size of antiderivative = 0.94, 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, 4731, 12, 1809, 1267, 459, 364} \[ \frac {3 d^2 e (f x)^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{f (m+1)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (m+7)}-\frac {b c (f x)^{m+2} \left (\frac {e \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^6 (m+3)^2 (m+5)^2 (m+7)^2}+\frac {d^3}{m^2+3 m+2}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{f^2}+\frac {b e \sqrt {1-c^2 x^2} (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^5 f^2 (m+3)^2 (m+5)^2 (m+7)^2}+\frac {b e^2 \sqrt {1-c^2 x^2} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^3 f^4 (m+5)^2 (m+7)^2}+\frac {b e^3 \sqrt {1-c^2 x^2} (f x)^{m+6}}{c f^6 (m+7)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 364
Rule 459
Rule 1267
Rule 1809
Rule 4731
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-(b c) \int \frac {(f x)^{1+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 )}{f \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {(b c) \int \frac {(f x)^{1+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}{f}\\ &=\frac {b e^3 (f x)^{6+m} \sqrt {1-c^2 x^2}}{c f^6 (7+m)^2}+\frac {d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}+\frac {b \int \frac {(f x)^{1+m} \left (-\frac {c^2 d^3 (7+m)}{1+m}-\frac {3 c^2 d^2 e (7+m) x^2}{3+m}-\frac {e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) x^4}{(5+m) (7+m)}\right )}{\sqrt {1-c^2 x^2}} \, dx}{c f (7+m)}\\ &=\frac {b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \sqrt {1-c^2 x^2}}{c^3 f^4 (5+m)^2 (7+m)^2}+\frac {b e^3 (f x)^{6+m} \sqrt {1-c^2 x^2}}{c f^6 (7+m)^2}+\frac {d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {b \int \frac {(f x)^{1+m} \left (\frac {c^4 d^3 (5+m) (7+m)}{1+m}+\frac {e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt {1-c^2 x^2}} \, dx}{c^3 f (5+m) (7+m)}\\ &=\frac {b e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) (f x)^{2+m} \sqrt {1-c^2 x^2}}{c^5 f^2 (3+m)^2 (5+m)^2 (7+m)^2}+\frac {b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \sqrt {1-c^2 x^2}}{c^3 f^4 (5+m)^2 (7+m)^2}+\frac {b e^3 (f x)^{6+m} \sqrt {1-c^2 x^2}}{c f^6 (7+m)^2}+\frac {d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b \left (\frac {c^6 d^3}{1+m}+\frac {e (2+m) \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right )}{(3+m)^2 (5+m)^2 (7+m)^2}\right )\right ) \int \frac {(f x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{c^5 f}\\ &=\frac {b e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) (f x)^{2+m} \sqrt {1-c^2 x^2}}{c^5 f^2 (3+m)^2 (5+m)^2 (7+m)^2}+\frac {b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \sqrt {1-c^2 x^2}}{c^3 f^4 (5+m)^2 (7+m)^2}+\frac {b e^3 (f x)^{6+m} \sqrt {1-c^2 x^2}}{c f^6 (7+m)^2}+\frac {d^3 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \sin ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {b \left (\frac {c^6 d^3}{1+m}+\frac {e (2+m) \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right )}{(3+m)^2 (5+m)^2 (7+m)^2}\right ) (f x)^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{c^5 f^2 (2+m)}\\ \end {align*}
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Mathematica [F] time = 5.39, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} + {\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \arcsin \left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]
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 \[ \int {\left (e x^{2} + d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 44.16, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )\, dx \]
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 \[ \frac {a e^{3} f^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a d e^{2} f^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a d^{2} e f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d^{3}}{f {\left (m + 1\right )}} + \frac {{\left ({\left (b e^{3} f^{m} m^{3} + 9 \, b e^{3} f^{m} m^{2} + 23 \, b e^{3} f^{m} m + 15 \, b e^{3} f^{m}\right )} x^{7} + 3 \, {\left (b d e^{2} f^{m} m^{3} + 11 \, b d e^{2} f^{m} m^{2} + 31 \, b d e^{2} f^{m} m + 21 \, b d e^{2} f^{m}\right )} x^{5} + 3 \, {\left (b d^{2} e f^{m} m^{3} + 13 \, b d^{2} e f^{m} m^{2} + 47 \, b d^{2} e f^{m} m + 35 \, b d^{2} e f^{m}\right )} x^{3} + {\left (b d^{3} f^{m} m^{3} + 15 \, b d^{3} f^{m} m^{2} + 71 \, b d^{3} f^{m} m + 105 \, b d^{3} f^{m}\right )} x\right )} x^{m} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) - {\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} \int \frac {{\left ({\left (b c e^{3} f^{m} m^{3} + 9 \, b c e^{3} f^{m} m^{2} + 23 \, b c e^{3} f^{m} m + 15 \, b c e^{3} f^{m}\right )} x^{7} + 3 \, {\left (b c d e^{2} f^{m} m^{3} + 11 \, b c d e^{2} f^{m} m^{2} + 31 \, b c d e^{2} f^{m} m + 21 \, b c d e^{2} f^{m}\right )} x^{5} + 3 \, {\left (b c d^{2} e f^{m} m^{3} + 13 \, b c d^{2} e f^{m} m^{2} + 47 \, b c d^{2} e f^{m} m + 35 \, b c d^{2} e f^{m}\right )} x^{3} + {\left (b c d^{3} f^{m} m^{3} + 15 \, b c d^{3} f^{m} m^{2} + 71 \, b c d^{3} f^{m} m + 105 \, b c d^{3} f^{m}\right )} x\right )} \sqrt {c x + 1} \sqrt {-c x + 1} x^{m}}{m^{4} + 16 \, m^{3} - {\left (c^{2} m^{4} + 16 \, c^{2} m^{3} + 86 \, c^{2} m^{2} + 176 \, c^{2} m + 105 \, c^{2}\right )} x^{2} + 86 \, m^{2} + 176 \, m + 105}\,{d x}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]
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 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^3 \,d x \]
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 \[ \int \left (f x\right )^{m} \left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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