Optimal. Leaf size=190 \[ \frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \sqrt {1-c^2 x^2}}{5 c^5}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^{3/2}}{15 c^5}+\frac {b e^3 \left (1-c^2 x^2\right )^{5/2}}{25 c^5}-\frac {d^3 (a+b \text {ArcSin}(c x))}{x}+3 d^2 e x (a+b \text {ArcSin}(c x))+d e^2 x^3 (a+b \text {ArcSin}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {ArcSin}(c x))-b c d^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {276, 4815,
1813, 1634, 65, 214} \begin {gather*} -\frac {d^3 (a+b \text {ArcSin}(c x))}{x}+3 d^2 e x (a+b \text {ArcSin}(c x))+d e^2 x^3 (a+b \text {ArcSin}(c x))+\frac {1}{5} e^3 x^5 (a+b \text {ArcSin}(c x))-b c d^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-\frac {b e^2 \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+2 e\right )}{15 c^5}+\frac {b e^3 \left (1-c^2 x^2\right )^{5/2}}{25 c^5}+\frac {b e \sqrt {1-c^2 x^2} \left (15 c^4 d^2+5 c^2 d e+e^2\right )}{5 c^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 276
Rule 1634
Rule 1813
Rule 4815
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \sin ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {-d^3+3 d^2 e x^2+d e^2 x^4+\frac {e^3 x^6}{5}}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \sin ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {-d^3+3 d^2 e x+d e^2 x^2+\frac {e^3 x^3}{5}}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \sin ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \left (\frac {e \left (15 c^4 d^2+5 c^2 d e+e^2\right )}{5 c^4 \sqrt {1-c^2 x}}-\frac {d^3}{x \sqrt {1-c^2 x}}-\frac {e^2 \left (5 c^2 d+2 e\right ) \sqrt {1-c^2 x}}{5 c^4}+\frac {e^3 \left (1-c^2 x\right )^{3/2}}{5 c^4}\right ) \, dx,x,x^2\right )\\ &=\frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \sqrt {1-c^2 x^2}}{5 c^5}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^{3/2}}{15 c^5}+\frac {b e^3 \left (1-c^2 x^2\right )^{5/2}}{25 c^5}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \sin ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} \left (b c d^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \sqrt {1-c^2 x^2}}{5 c^5}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^{3/2}}{15 c^5}+\frac {b e^3 \left (1-c^2 x^2\right )^{5/2}}{25 c^5}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \sin ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b d^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c}\\ &=\frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \sqrt {1-c^2 x^2}}{5 c^5}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^{3/2}}{15 c^5}+\frac {b e^3 \left (1-c^2 x^2\right )^{5/2}}{25 c^5}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \sin ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-b c d^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 183, normalized size = 0.96 \begin {gather*} -\frac {a d^3}{x}+3 a d^2 e x+a d e^2 x^3+\frac {1}{5} a e^3 x^5+\frac {b e \sqrt {1-c^2 x^2} \left (8 e^2+2 c^2 e \left (25 d+2 e x^2\right )+c^4 \left (225 d^2+25 d e x^2+3 e^2 x^4\right )\right )}{75 c^5}+\frac {b \left (-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right ) \text {ArcSin}(c x)}{5 x}+b c d^3 \log (x)-b c d^3 \log \left (1+\sqrt {1-c^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 264, normalized size = 1.39
method | result | size |
derivativedivides | \(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \arcsin \left (c x \right ) c^{5} d^{2} e x +\arcsin \left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\arcsin \left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\arcsin \left (c x \right ) c^{5} d^{3}}{x}+3 c^{4} d^{2} e \sqrt {-c^{2} x^{2}+1}-c^{2} d \,e^{2} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-\frac {e^{3} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-c^{6} d^{3} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{6}}\right )\) | \(264\) |
default | \(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {b \left (3 \arcsin \left (c x \right ) c^{5} d^{2} e x +\arcsin \left (c x \right ) c^{5} d \,e^{2} x^{3}+\frac {\arcsin \left (c x \right ) e^{3} c^{5} x^{5}}{5}-\frac {\arcsin \left (c x \right ) c^{5} d^{3}}{x}+3 c^{4} d^{2} e \sqrt {-c^{2} x^{2}+1}-c^{2} d \,e^{2} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-\frac {e^{3} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-c^{6} d^{3} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{6}}\right )\) | \(264\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 239, normalized size = 1.26 \begin {gather*} \frac {1}{5} \, a x^{5} e^{3} + a d x^{3} e^{2} - {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b d^{3} + 3 \, a d^{2} x e + \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 e^{2} + \frac {3 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{2} e}{c} - \frac {a d^{3}}{x} + \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 e^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.32, size = 232, normalized size = 1.22 \begin {gather*} \frac {30 \, a c^{5} x^{6} e^{3} + 150 \, a c^{5} d x^{4} e^{2} - 75 \, b c^{6} d^{3} x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) + 75 \, b c^{6} d^{3} x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 450 \, a c^{5} d^{2} x^{2} e - 150 \, a c^{5} d^{3} + 30 \, {\left (b c^{5} x^{6} e^{3} + 5 \, b c^{5} d x^{4} e^{2} + 15 \, b c^{5} d^{2} x^{2} e - 5 \, b c^{5} d^{3}\right )} \arcsin \left (c x\right ) + 2 \, {\left (225 \, b c^{4} d^{2} x e + {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} e^{3} + 25 \, {\left (b c^{4} d x^{3} + 2 \, b c^{2} d x\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{150 \, c^{5} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.71, size = 272, normalized size = 1.43 \begin {gather*} - \frac {a d^{3}}{x} + 3 a d^{2} e x + a d e^{2} x^{3} + \frac {a e^{3} x^{5}}{5} + b c d^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - b c d e^{2} \left (\begin {cases} - \frac {x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c^{2}} - \frac {2 \sqrt {- c^{2} x^{2} + 1}}{3 c^{4}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - \frac {b c e^{3} \left (\begin {cases} - \frac {x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c^{2}} - \frac {4 x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{4}} - \frac {8 \sqrt {- c^{2} x^{2} + 1}}{15 c^{6}} & \text {for}\: c \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right )}{5} - \frac {b d^{3} \operatorname {asin}{\left (c x \right )}}{x} + 3 b d^{2} e \left (\begin {cases} 0 & \text {for}\: c = 0 \\x \operatorname {asin}{\left (c x \right )} + \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) + b d e^{2} x^{3} \operatorname {asin}{\left (c x \right )} + \frac {b e^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 10769 vs.
\(2 (174) = 348\).
time = 11.48, size = 10769, normalized size = 56.68 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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