Optimal. Leaf size=596 \[ -\frac {b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^6 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}-\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) (f x)^{3+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^4 f^3 (4+m) (5+m) (6+m) (7+m)}-\frac {b e^3 (f x)^{5+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2 f^5 (6+m) (7+m)}+\frac {d^3 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (7+m)}+\frac {b \left (\frac {c^6 d^3 (2+m) (4+m) (6+m)}{1+m}+\frac {e (1+m) \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{(3+m) (5+m) (7+m)}\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{c^6 f (1+m) (2+m) (4+m) (6+m)} \]
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Rubi [A]
time = 1.57, antiderivative size = 576, normalized size of antiderivative = 0.97, number of steps
used = 5, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {276, 6436,
1823, 1281, 470, 371} \begin {gather*} \frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}-\frac {b e^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)}-\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^4 f^3 (m+4) (m+5) (m+6) (m+7)}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^6 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (f x)^{m+1} \left (\frac {e \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^6 (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+\frac {d^3}{(m+1)^2}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 371
Rule 470
Rule 1281
Rule 1823
Rule 6436
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {d^3 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (7+m)}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(f x)^m \left (\frac {d^3}{1+m}+\frac {3 d^2 e x^2}{3+m}+\frac {3 d e^2 x^4}{5+m}+\frac {e^3 x^6}{7+m}\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b e^3 (f x)^{5+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2 f^5 (6+m) (7+m)}+\frac {d^3 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(f x)^m \left (-\frac {c^2 d^3 (6+m)}{1+m}-\frac {3 c^2 d^2 e (6+m) x^2}{3+m}-\frac {e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x^4}{(5+m) (7+m)}\right )}{\sqrt {1-c^2 x^2}} \, dx}{c^2 (6+m)}\\ &=-\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) (f x)^{3+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^4 f^3 (4+m) (5+m) (6+m) (7+m)}-\frac {b e^3 (f x)^{5+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2 f^5 (6+m) (7+m)}+\frac {d^3 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (7+m)}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(f x)^m \left (\frac {c^4 d^3 (4+m) (6+m)}{1+m}+\frac {e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt {1-c^2 x^2}} \, dx}{c^4 (4+m) (6+m)}\\ &=-\frac {b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^6 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}-\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) (f x)^{3+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^4 f^3 (4+m) (5+m) (6+m) (7+m)}-\frac {b e^3 (f x)^{5+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2 f^5 (6+m) (7+m)}+\frac {d^3 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (7+m)}+\frac {\left (b \left (\frac {c^4 d^3 (4+m) (6+m)}{1+m}+\frac {e (1+m) \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{c^2 (2+m) (3+m) (5+m) (7+m)}\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(f x)^m}{\sqrt {1-c^2 x^2}} \, dx}{c^4 (4+m) (6+m)}\\ &=-\frac {b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^6 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}-\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) (f x)^{3+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^4 f^3 (4+m) (5+m) (6+m) (7+m)}-\frac {b e^3 (f x)^{5+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2 f^5 (6+m) (7+m)}+\frac {d^3 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (7+m)}+\frac {b \left (\frac {d^3}{(1+m)^2}+\frac {e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{c^6 (2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{f}\\ \end {align*}
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Mathematica [F]
time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 1.36, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )\, dx\]
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (f x\right )^{m} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \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 {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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