Optimal. Leaf size=585 \[ \frac {b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac {b \left (\frac {c^6 d^3 (2+m) (4+m) (6+m)}{1+m}+\frac {e (1+m) \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{(3+m) (5+m) (7+m)}\right ) x (f x)^{1+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{c^5 f (1+m) (2+m) (4+m) (6+m) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.57, antiderivative size = 566, normalized size of antiderivative = 0.97, number of steps
used = 6, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {276, 5347,
1823, 1281, 470, 372, 371} \begin {gather*} \frac {d^3 (f x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (m+7)}+\frac {b e^3 x \sqrt {c^2 x^2-1} (f x)^{m+5}}{c f^5 (m+6) (m+7) \sqrt {c^2 x^2}}+\frac {b e^2 x \sqrt {c^2 x^2-1} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^3 f^3 (m+4) (m+5) (m+6) (m+7) \sqrt {c^2 x^2}}+\frac {b c x \sqrt {1-c^2 x^2} (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 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1}}+\frac {b e x \sqrt {c^2 x^2-1} (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^5 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \sqrt {c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 276
Rule 371
Rule 372
Rule 470
Rule 1281
Rule 1823
Rule 5347
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac {(b c x) \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}{\sqrt {c^2 x^2}}\\ &=\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac {(b x) \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 (6+m) \sqrt {c^2 x^2}}\\ &=\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac {(b x) \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^3 (4+m) (6+m) \sqrt {c^2 x^2}}\\ &=\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 ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \csc ^{-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 ) x\right ) \int \frac {(f x)^m}{\sqrt {-1+c^2 x^2}} \, dx}{c^3 (4+m) (6+m) \sqrt {c^2 x^2}}\\ &=\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 ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \csc ^{-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 ) x \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^m}{\sqrt {1-c^2 x^2}} \, dx}{c^3 (4+m) (6+m) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}\\ &=\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 ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac {b \left (\frac {c^6 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 )}{(2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}\right ) x (f x)^{1+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{c^5 f \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{3} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________