Optimal. Leaf size=568 \[ \frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 f \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 f x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {5 d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.57, antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5972, 5975,
5898, 5896, 5893, 30, 74, 14, 267, 5915, 41, 200} \begin {gather*} \frac {1}{6} d^2 f x (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 g (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 f \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 41
Rule 74
Rule 200
Rule 267
Rule 5893
Rule 5896
Rule 5898
Rule 5915
Rule 5972
Rule 5975
Rubi steps
\begin {align*} \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} (f+g x) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (f (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+g x (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (d^2 f \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 g \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{6} d^2 f x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right )^2 \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^3 \, dx}{7 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 f \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{24} d^2 f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 f x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d^2 g \sqrt {d-c^2 d x^2}\right ) \int \left (-1+3 c^2 x^2-3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 f \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 f x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {\left (5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int \left (-x+c^2 x^3\right ) \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {25 b c d^2 f x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d^2 f x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 f \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 f x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {5 d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 4.19, size = 644, normalized size = 1.13 \begin {gather*} \frac {d^2 \left (8400 a \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )-882000 a c \sqrt {d} f \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+78400 b g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )-352800 b c f \sqrt {d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+44100 b c f \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )-1568 b g \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)-25 \cosh \left (3 \cosh ^{-1}(c x)\right )-9 \cosh \left (5 \cosh ^{-1}(c x)\right )+75 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+45 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right )+1225 b c f \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )+4 b g \sqrt {d-c^2 d x^2} \left (55125 c x-55125 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)-1225 \cosh \left (3 \cosh ^{-1}(c x)\right )-1323 \cosh \left (5 \cosh ^{-1}(c x)\right )-225 \cosh \left (7 \cosh ^{-1}(c x)\right )+3675 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+6615 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )+1575 \cosh ^{-1}(c x) \sinh \left (7 \cosh ^{-1}(c x)\right )\right )\right )}{2822400 c^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1855\) vs.
\(2(488)=976\).
time = 7.12, size = 1856, normalized size = 3.27
method | result | size |
default | \(\text {Expression too large to display}\) | \(1856\) |
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 (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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