Optimal. Leaf size=416 \[ \frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {b c^2 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{75 d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{75 d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.39, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {270, 5346,
12, 485, 594, 597, 538, 438, 437, 435, 432, 430} \begin {gather*} -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {b x \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{75 d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2} E\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{75 d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {4 b c \sqrt {c^2 x^2-1} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}+\frac {b c \sqrt {c^2 x^2-1} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 430
Rule 432
Rule 435
Rule 437
Rule 438
Rule 485
Rule 538
Rule 594
Rule 597
Rule 5346
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^6} \, dx &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {(b c x) \int -\frac {\left (d+e x^2\right )^{5/2}}{5 d x^6 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2}}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{5 d \sqrt {c^2 x^2}}\\ &=\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-4 d \left (c^2 d+2 e\right )-e \left (c^2 d+5 e\right ) x^2\right )}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{25 d \sqrt {c^2 x^2}}\\ &=\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {(b c x) \int \frac {d \left (8 c^4 d^2+23 c^2 d e+23 e^2\right )+e \left (4 c^4 d^2+11 c^2 d e+15 e^2\right ) x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d \sqrt {c^2 x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {(b c x) \int \frac {d e \left (4 c^4 d^2+11 c^2 d e+15 e^2\right )-c^2 d e \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d^2 \sqrt {c^2 x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {\left (b c \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d \sqrt {c^2 x^2}}-\frac {\left (b c^3 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{75 d \sqrt {c^2 x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {\left (b c^3 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{75 d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (b c \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{75 d \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {\left (b c^3 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{75 d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{75 d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {c^2 x^2}}+\frac {4 b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}-\frac {b c^2 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{75 d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{75 d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.06, size = 303, normalized size = 0.73 \begin {gather*} \frac {\sqrt {d+e x^2} \left (-15 a \left (d+e x^2\right )^2+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (23 e^2 x^4+d e x^2 \left (11+23 c^2 x^2\right )+d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (d+e x^2\right )^2 \sec ^{-1}(c x)\right )}{75 d x^5}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (8 c^6 d^3+27 c^4 d^2 e+34 c^2 d e^2+15 e^3\right ) F\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )\right )}{75 \sqrt {-c^2} d \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.81, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{x^{6}}\, dx\]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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