Optimal. Leaf size=152 \[ \frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \left (1-c^2 x^2\right )^{5/2} \left (7 c^2 d+15 e\right )}{175 c^7}-\frac {b \left (1-c^2 x^2\right )^{3/2} \left (14 c^2 d+15 e\right )}{105 c^7}+\frac {b \sqrt {1-c^2 x^2} \left (7 c^2 d+5 e\right )}{35 c^7}-\frac {b e \left (1-c^2 x^2\right )^{7/2}}{49 c^7} \]
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Rubi [A] time = 0.15, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 4731, 12, 446, 77} \[ \frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \left (1-c^2 x^2\right )^{5/2} \left (7 c^2 d+15 e\right )}{175 c^7}-\frac {b \left (1-c^2 x^2\right )^{3/2} \left (14 c^2 d+15 e\right )}{105 c^7}+\frac {b \sqrt {1-c^2 x^2} \left (7 c^2 d+5 e\right )}{35 c^7}-\frac {b e \left (1-c^2 x^2\right )^{7/2}}{49 c^7} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 77
Rule 446
Rule 4731
Rubi steps
\begin {align*} \int x^4 \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x^5 \left (7 d+5 e x^2\right )}{35 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{35} (b c) \int \frac {x^5 \left (7 d+5 e x^2\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{70} (b c) \operatorname {Subst}\left (\int \frac {x^2 (7 d+5 e x)}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{70} (b c) \operatorname {Subst}\left (\int \left (\frac {7 c^2 d+5 e}{c^6 \sqrt {1-c^2 x}}+\frac {\left (-14 c^2 d-15 e\right ) \sqrt {1-c^2 x}}{c^6}+\frac {\left (7 c^2 d+15 e\right ) \left (1-c^2 x\right )^{3/2}}{c^6}-\frac {5 e \left (1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )\\ &=\frac {b \left (7 c^2 d+5 e\right ) \sqrt {1-c^2 x^2}}{35 c^7}-\frac {b \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{3/2}}{105 c^7}+\frac {b \left (7 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}-\frac {b e \left (1-c^2 x^2\right )^{7/2}}{49 c^7}+\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 115, normalized size = 0.76 \[ \frac {105 a x^5 \left (7 d+5 e x^2\right )+\frac {b \sqrt {1-c^2 x^2} \left (3 c^6 \left (49 d x^4+25 e x^6\right )+2 c^4 \left (98 d x^2+45 e x^4\right )+8 c^2 \left (49 d+15 e x^2\right )+240 e\right )}{c^7}+105 b x^5 \sin ^{-1}(c x) \left (7 d+5 e x^2\right )}{3675} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 128, normalized size = 0.84 \[ \frac {525 \, a c^{7} e x^{7} + 735 \, a c^{7} d x^{5} + 105 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5}\right )} \arcsin \left (c x\right ) + {\left (75 \, b c^{6} e x^{6} + 3 \, {\left (49 \, b c^{6} d + 30 \, b c^{4} e\right )} x^{4} + 392 \, b c^{2} d + 4 \, {\left (49 \, b c^{4} d + 30 \, b c^{2} e\right )} x^{2} + 240 \, b e\right )} \sqrt {-c^{2} x^{2} + 1}}{3675 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 325, normalized size = 2.14 \[ \frac {1}{7} \, a x^{7} e + \frac {1}{5} \, a d x^{5} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b x \arcsin \left (c x\right ) e}{7 \, c^{6}} + \frac {b d x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b x \arcsin \left (c x\right ) e}{7 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d}{25 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e}{7 \, c^{6}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d}{15 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e}{49 \, c^{7}} + \frac {b x \arcsin \left (c x\right ) e}{7 \, c^{6}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d}{5 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e}{35 \, c^{7}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e}{7 \, c^{7}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e}{7 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 201, normalized size = 1.32 \[ \frac {\frac {a \left (\frac {1}{7} e \,c^{7} x^{7}+\frac {1}{5} c^{7} x^{5} d \right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e \,c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{7} x^{5} d}{5}-\frac {e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-\frac {c^{2} d \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{2}}}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 183, normalized size = 1.20 \[ \frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d + \frac {1}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e \]
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 \[ \int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.69, size = 223, normalized size = 1.47 \[ \begin {cases} \frac {a d x^{5}}{5} + \frac {a e x^{7}}{7} + \frac {b d x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b e x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {b d x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b e x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} + \frac {4 b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {6 b e x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} + \frac {8 b d \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + \frac {8 b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} + \frac {16 b e \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\a \left (\frac {d x^{5}}{5} + \frac {e x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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