Optimal. Leaf size=226 \[ \frac {b c \sqrt {1-c^2 x^2}}{15 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {2 b c \left (3 c^2 d+2 e\right ) \sqrt {1-c^2 x^2}}{15 d^2 \left (c^2 d+e\right )^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {ArcSin}(c x))}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x (a+b \text {ArcSin}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x (a+b \text {ArcSin}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {8 b \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{15 d^3 \sqrt {e}} \]
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Rubi [A]
time = 0.58, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {198, 197,
4755, 12, 6847, 963, 79, 65, 223, 209} \begin {gather*} \frac {8 x (a+b \text {ArcSin}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {ArcSin}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {ArcSin}(c x))}{5 d \left (d+e x^2\right )^{5/2}}+\frac {8 b \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{15 d^3 \sqrt {e}}+\frac {2 b c \sqrt {1-c^2 x^2} \left (3 c^2 d+2 e\right )}{15 d^2 \left (c^2 d+e\right )^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {1-c^2 x^2}}{15 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 79
Rule 197
Rule 198
Rule 209
Rule 223
Rule 963
Rule 4755
Rule 6847
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^3 \sqrt {d+e x^2}}-(b c) \int \frac {x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{15 d^3 \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}} \, dx\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^3 \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}} \, dx}{15 d^3}\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^3 \sqrt {d+e x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {1-c^2 x} (d+e x)^{5/2}} \, dx,x,x^2\right )}{30 d^3}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{15 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^3 \sqrt {d+e x^2}}+\frac {(b c) \text {Subst}\left (\int \frac {-3 d \left (7 c^2 d+6 e\right )-12 e \left (c^2 d+e\right ) x}{\sqrt {1-c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{45 d^3 \left (c^2 d+e\right )}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{15 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {2 b c \left (3 c^2 d+2 e\right ) \sqrt {1-c^2 x^2}}{15 d^2 \left (c^2 d+e\right )^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^3 \sqrt {d+e x^2}}-\frac {(4 b c) \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{15 d^3}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{15 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {2 b c \left (3 c^2 d+2 e\right ) \sqrt {1-c^2 x^2}}{15 d^2 \left (c^2 d+e\right )^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^3 \sqrt {d+e x^2}}+\frac {(8 b) \text {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}-\frac {e x^2}{c^2}}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{15 c d^3}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{15 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {2 b c \left (3 c^2 d+2 e\right ) \sqrt {1-c^2 x^2}}{15 d^2 \left (c^2 d+e\right )^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^3 \sqrt {d+e x^2}}+\frac {(8 b) \text {Subst}\left (\int \frac {1}{1+\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {1-c^2 x^2}}{\sqrt {d+e x^2}}\right )}{15 c d^3}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{15 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {2 b c \left (3 c^2 d+2 e\right ) \sqrt {1-c^2 x^2}}{15 d^2 \left (c^2 d+e\right )^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x \left (a+b \sin ^{-1}(c x)\right )}{15 d^3 \sqrt {d+e x^2}}+\frac {8 b \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{15 d^3 \sqrt {e}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 0.27, size = 188, normalized size = 0.83 \begin {gather*} \frac {a x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )+\frac {b c d \sqrt {1-c^2 x^2} \left (d+e x^2\right ) \left (e \left (5 d+4 e x^2\right )+c^2 d \left (7 d+6 e x^2\right )\right )}{\left (c^2 d+e\right )^2}-4 b c x^2 \left (d+e x^2\right )^2 \sqrt {1+\frac {e x^2}{d}} F_1\left (1;\frac {1}{2},\frac {1}{2};2;c^2 x^2,-\frac {e x^2}{d}\right )+b x \left (15 d^2+20 d e x^2+8 e^2 x^4\right ) \text {ArcSin}(c x)}{15 d^3 \left (d+e x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.82, size = 0, normalized size = 0.00 \[\int \frac {a +b \arcsin \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 660 vs.
\(2 (199) = 398\).
time = 3.09, size = 660, normalized size = 2.92 \begin {gather*} \frac {4 \, {\left (b c^{4} d^{5} + b x^{6} e^{5} + {\left (2 \, b c^{2} d x^{6} + 3 \, b d x^{4}\right )} e^{4} + {\left (b c^{4} d^{2} x^{6} + 6 \, b c^{2} d^{2} x^{4} + 3 \, b d^{2} x^{2}\right )} e^{3} + {\left (3 \, b c^{4} d^{3} x^{4} + 6 \, b c^{2} d^{3} x^{2} + b d^{3}\right )} e^{2} + {\left (3 \, b c^{4} d^{4} x^{2} + 2 \, b c^{2} d^{4}\right )} e\right )} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} {\left (c^{2} d + {\left (2 \, c^{2} x^{2} - 1\right )} e\right )} \sqrt {x^{2} e + d} e^{\frac {1}{2}}}{2 \, {\left ({\left (c^{3} x^{4} - c x^{2}\right )} e^{2} + {\left (c^{3} d x^{2} - c d\right )} e\right )}}\right ) e^{\frac {1}{2}} + {\left (15 \, a c^{4} d^{4} x e + 8 \, a x^{5} e^{5} + {\left (15 \, b c^{4} d^{4} x e + 8 \, b x^{5} e^{5} + 4 \, {\left (4 \, b c^{2} d x^{5} + 5 \, b d x^{3}\right )} e^{4} + {\left (8 \, b c^{4} d^{2} x^{5} + 40 \, b c^{2} d^{2} x^{3} + 15 \, b d^{2} x\right )} e^{3} + 10 \, {\left (2 \, b c^{4} d^{3} x^{3} + 3 \, b c^{2} d^{3} x\right )} e^{2}\right )} \arcsin \left (c x\right ) + 4 \, {\left (4 \, a c^{2} d x^{5} + 5 \, a d x^{3}\right )} e^{4} + {\left (8 \, a c^{4} d^{2} x^{5} + 40 \, a c^{2} d^{2} x^{3} + 15 \, a d^{2} x\right )} e^{3} + 10 \, {\left (2 \, a c^{4} d^{3} x^{3} + 3 \, a c^{2} d^{3} x\right )} e^{2} + {\left (7 \, b c^{3} d^{4} e + 4 \, b c d x^{4} e^{4} + 3 \, {\left (2 \, b c^{3} d^{2} x^{4} + 3 \, b c d^{2} x^{2}\right )} e^{3} + {\left (13 \, b c^{3} d^{3} x^{2} + 5 \, b c d^{3}\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {x^{2} e + d}}{15 \, {\left (c^{4} d^{8} e + d^{3} x^{6} e^{6} + {\left (2 \, c^{2} d^{4} x^{6} + 3 \, d^{4} x^{4}\right )} e^{5} + {\left (c^{4} d^{5} x^{6} + 6 \, c^{2} d^{5} x^{4} + 3 \, d^{5} x^{2}\right )} e^{4} + {\left (3 \, c^{4} d^{6} x^{4} + 6 \, c^{2} d^{6} x^{2} + d^{6}\right )} e^{3} + {\left (3 \, c^{4} d^{7} x^{2} + 2 \, c^{2} d^{7}\right )} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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