Optimal. Leaf size=225 \[ d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {3 b e^2 \left (1-c^2 x^2\right )^{5/2} \left (7 c^2 d+5 e\right )}{175 c^7}-\frac {b e^3 \left (1-c^2 x^2\right )^{7/2}}{49 c^7}-\frac {b e \left (1-c^2 x^2\right )^{3/2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7}+\frac {b \sqrt {1-c^2 x^2} \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right )}{35 c^7} \]
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Rubi [A] time = 0.25, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {194, 4665, 12, 1799, 1850} \[ d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 x \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {b e \left (1-c^2 x^2\right )^{3/2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7}+\frac {b \sqrt {1-c^2 x^2} \left (35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2+5 e^3\right )}{35 c^7}+\frac {3 b e^2 \left (1-c^2 x^2\right )^{5/2} \left (7 c^2 d+5 e\right )}{175 c^7}-\frac {b e^3 \left (1-c^2 x^2\right )^{7/2}}{49 c^7} \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 1799
Rule 1850
Rule 4665
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{35 \sqrt {1-c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{35} (b c) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{70} (b c) \operatorname {Subst}\left (\int \frac {35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{70} (b c) \operatorname {Subst}\left (\int \left (\frac {35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3}{c^6 \sqrt {1-c^2 x}}-\frac {e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt {1-c^2 x}}{c^6}+\frac {3 e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x\right )^{3/2}}{c^6}-\frac {5 e^3 \left (1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )\\ &=\frac {b \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right ) \sqrt {1-c^2 x^2}}{35 c^7}-\frac {b e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{105 c^7}+\frac {3 b e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}-\frac {b e^3 \left (1-c^2 x^2\right )^{7/2}}{49 c^7}+d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.24, size = 187, normalized size = 0.83 \[ \frac {105 a x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )+\frac {b \sqrt {1-c^2 x^2} \left (c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+24 c^2 e^2 \left (49 d+5 e x^2\right )+240 e^3\right )}{c^7}+105 b x \sin ^{-1}(c x) \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{3675} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 229, normalized size = 1.02 \[ \frac {525 \, a c^{7} e^{3} x^{7} + 2205 \, a c^{7} d e^{2} x^{5} + 3675 \, a c^{7} d^{2} e x^{3} + 3675 \, a c^{7} d^{3} x + 105 \, {\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \arcsin \left (c x\right ) + {\left (75 \, b c^{6} e^{3} x^{6} + 3675 \, b c^{6} d^{3} + 2450 \, b c^{4} d^{2} e + 1176 \, b c^{2} d e^{2} + 9 \, {\left (49 \, b c^{6} d e^{2} + 10 \, b c^{4} e^{3}\right )} x^{4} + 240 \, b e^{3} + {\left (1225 \, b c^{6} d^{2} e + 588 \, b c^{4} d e^{2} + 120 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{3675 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.98, size = 469, normalized size = 2.08 \[ \frac {1}{7} \, a x^{7} e^{3} + \frac {3}{5} \, a d x^{5} e^{2} + a d^{2} x^{3} e + b d^{3} x \arcsin \left (c x\right ) + a d^{3} x + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right ) e}{c^{2}} + \frac {b d^{2} x \arcsin \left (c x\right ) e}{c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{3}}{c} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} e}{3 \, c^{3}} + \frac {6 \, {\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} e}{c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b x \arcsin \left (c x\right ) e^{3}}{7 \, c^{6}} + \frac {3 \, b d x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e^{2}}{25 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b x \arcsin \left (c x\right ) e^{3}}{7 \, c^{6}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e^{2}}{5 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e^{3}}{7 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e^{3}}{49 \, c^{7}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2}}{5 \, c^{5}} + \frac {b x \arcsin \left (c x\right ) e^{3}}{7 \, c^{6}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{3}}{35 \, c^{7}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{3}}{7 \, c^{7}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{3}}{7 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 325, normalized size = 1.44 \[ \frac {\frac {a \left (\frac {1}{7} e^{3} c^{7} x^{7}+\frac {3}{5} c^{7} d \,e^{2} x^{5}+c^{7} d^{2} e \,x^{3}+d^{3} c^{7} x \right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{7} d \,e^{2} x^{5}}{5}+\arcsin \left (c x \right ) c^{7} d^{2} e \,x^{3}+\arcsin \left (c x \right ) d^{3} c^{7} x -\frac {e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-\frac {3 c^{2} d \,e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-c^{4} d^{2} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}\right )}{c^{6}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 292, normalized size = 1.30 \[ \frac {1}{7} \, a e^{3} x^{7} + \frac {3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} e + \frac {1}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e^{2} + \frac {1}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{3} + a d^{3} x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3}}{c} \]
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 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.28, size = 389, normalized size = 1.73 \[ \begin {cases} a d^{3} x + a d^{2} e x^{3} + \frac {3 a d e^{2} x^{5}}{5} + \frac {a e^{3} x^{7}}{7} + b d^{3} x \operatorname {asin}{\left (c x \right )} + b d^{2} e x^{3} \operatorname {asin}{\left (c x \right )} + \frac {3 b d e^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b e^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {b d^{3} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d^{2} e x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {3 b d e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b e^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} + \frac {2 b d^{2} e \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + \frac {4 b d e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{3}} + \frac {6 b e^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} + \frac {8 b d e^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{5}} + \frac {8 b e^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} + \frac {16 b e^{3} \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + d^{2} e x^{3} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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