Optimal. Leaf size=568 \[ \frac{1}{6} d^2 f x (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{24} d^2 f x (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 g (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac{5 b c^3 d^2 f x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}-\frac{25 b c d^2 f x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 f \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{36 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{c x-1} \sqrt{c x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.904908, antiderivative size = 568, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {5836, 5822, 5685, 5683, 5676, 30, 14, 261, 5718, 194} \[ \frac{1}{6} d^2 f x (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{5}{24} d^2 f x (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 g (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac{5 b c^3 d^2 f x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}-\frac{25 b c d^2 f x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 f \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{36 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5836
Rule 5822
Rule 5685
Rule 5683
Rule 5676
Rule 30
Rule 14
Rule 261
Rule 5718
Rule 194
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*}
Mathematica [A] time = 6.30286, size = 644, normalized size = 1.13 \[ \frac{d^2 \left (8400 a \sqrt{\frac{c x-1}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2} \left (7 c^2 f x \left (8 c^4 x^4-26 c^2 x^2+33\right )+48 g \left (c^2 x^2-1\right )^3\right )-882000 a c \sqrt{d} f \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-352800 b c f \sqrt{d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+44100 b c f \sqrt{d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+1225 b c f \sqrt{d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )+78400 b g \sqrt{d-c^2 d x^2} \left (9 c x+12 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )-1568 b g \sqrt{d-c^2 d x^2} \left (450 c x-450 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-25 \cosh \left (3 \cosh ^{-1}(c x)\right )-9 \cosh \left (5 \cosh ^{-1}(c x)\right )+75 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+45 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right )+4 b g \sqrt{d-c^2 d x^2} \left (55125 c x-55125 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-1225 \cosh \left (3 \cosh ^{-1}(c x)\right )-1323 \cosh \left (5 \cosh ^{-1}(c x)\right )-225 \cosh \left (7 \cosh ^{-1}(c x)\right )+3675 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+6615 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )+1575 \cosh ^{-1}(c x) \sinh \left (7 \cosh ^{-1}(c x)\right )\right )\right )}{2822400 c^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.465, size = 877, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a c^{4} d^{2} g x^{5} + a c^{4} d^{2} f x^{4} - 2 \, a c^{2} d^{2} g x^{3} - 2 \, a c^{2} d^{2} f x^{2} + a d^{2} g x + a d^{2} f +{\left (b c^{4} d^{2} g x^{5} + b c^{4} d^{2} f x^{4} - 2 \, b c^{2} d^{2} g x^{3} - 2 \, b c^{2} d^{2} f x^{2} + b d^{2} g x + b d^{2} f\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]