Optimal. Leaf size=558 \[ \frac {d^3 (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (m+7)}+\frac {b e^3 \left (1-c^2 x^2\right ) (f x)^{m+6}}{c f^6 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^2 \left (1-c^2 x^2\right ) (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^3 f^4 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \left (1-c^2 x^2\right ) (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^5 f^2 (m+3)^2 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \sqrt {1-c^2 x^2} (f x)^{m+2} \left (\frac {c^6 d^3 (m+3) (m+5) (m+7)}{m+1}+\frac {e (m+2) \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{(m+3) (m+5) (m+7)}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{c^5 f^2 (m+2) (m+3) (m+5) (m+7) \sqrt {c x-1} \sqrt {c x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.81, antiderivative size = 529, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {270, 5790, 12, 1610, 1809, 1267, 459, 365, 364} \[ \frac {3 d^2 e (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (m+7)}-\frac {b c \sqrt {1-c^2 x^2} (f x)^{m+2} \left (\frac {e \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^6 (m+3)^2 (m+5)^2 (m+7)^2}+\frac {d^3}{m^2+3 m+2}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{f^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \left (1-c^2 x^2\right ) (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^5 f^2 (m+3)^2 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^2 \left (1-c^2 x^2\right ) (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^3 f^4 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^3 \left (1-c^2 x^2\right ) (f x)^{m+6}}{c f^6 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 270
Rule 364
Rule 365
Rule 459
Rule 1267
Rule 1610
Rule 1809
Rule 5790
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-(b c) \int \frac {(f x)^{1+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 )}{f \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {(b c) \int \frac {(f x)^{1+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 x} \sqrt {1+c x}} \, dx}{f}\\ &=\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+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}{f \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{c f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {c^2 d^3 (7+m)}{1+m}+\frac {3 c^2 d^2 e (7+m) x^2}{3+m}+\frac {e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) x^4}{(5+m) (7+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c f (7+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \left (1-c^2 x^2\right )}{c^3 f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{c f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {c^4 d^3 (5+m) (7+m)}{1+m}+\frac {e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c^3 f (5+m) (7+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{c^5 f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \left (1-c^2 x^2\right )}{c^3 f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{c f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b \left (\frac {c^4 d^3 (5+m) (7+m)}{1+m}+\frac {e (2+m) \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right )}{c^2 (3+m)^2 (5+m) (7+m)}\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {-1+c^2 x^2}} \, dx}{c^3 f (5+m) (7+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{c^5 f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \left (1-c^2 x^2\right )}{c^3 f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{c f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {\left (b \left (\frac {c^4 d^3 (5+m) (7+m)}{1+m}+\frac {e (2+m) \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right )}{c^2 (3+m)^2 (5+m) (7+m)}\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{c^3 f (5+m) (7+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{c^5 f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \left (1-c^2 x^2\right )}{c^3 f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{c f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^7 (7+m)}-\frac {b \left (\frac {c^6 d^3}{2+3 m+m^2}+\frac {e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right )}{(3+m)^2 (5+m)^2 (7+m)^2}\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{c^5 f^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.42, size = 397, normalized size = 0.71 \[ x (f x)^m \left (\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{m+1}+\frac {3 d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )}{m+3}+\frac {3 d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )}{m+5}+\frac {e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )}{m+7}-\frac {b c d^3 x \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c d^2 e x^3 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};c^2 x^2\right )}{\left (m^2+7 m+12\right ) \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c d e^2 x^5 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+6}{2};\frac {m+8}{2};c^2 x^2\right )}{(m+5) (m+6) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c e^3 x^7 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m}{2}+4;\frac {m}{2}+5;c^2 x^2\right )}{(m+7) (m+8) \sqrt {c x-1} \sqrt {c x+1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} + {\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \operatorname {arcosh}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{3} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a e^{3} f^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a d e^{2} f^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a d^{2} e f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d^{3}}{f {\left (m + 1\right )}} + \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b e^{3} f^{m} x^{7} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} b d e^{2} f^{m} x^{5} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} b d^{2} e f^{m} x^{3} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b d^{3} f^{m} x\right )} x^{m} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} + \int \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b c e^{3} f^{m} x^{7} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} b c d e^{2} f^{m} x^{5} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} b c d^{2} e f^{m} x^{3} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b c d^{3} f^{m} x\right )} x^{m}}{{\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c^{3} x^{3} - {\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c x + {\left ({\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c^{2} x^{2} - m^{4} - 16 \, m^{3} - 86 \, m^{2} - 176 \, m - 105\right )} \sqrt {c x + 1} \sqrt {c x - 1}}\,{d x} - \int \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b c^{2} e^{3} f^{m} x^{8} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} b c^{2} d e^{2} f^{m} x^{6} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} b c^{2} d^{2} e f^{m} x^{4} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b c^{2} d^{3} f^{m} x^{2}\right )} x^{m}}{{\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c^{2} x^{2} - m^{4} - 16 \, m^{3} - 86 \, m^{2} - 176 \, m - 105}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________