Optimal. Leaf size=684 \[ \frac {b \left (1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+360 e^3 i+588 c^2 e \left (e^2 g+3 d e h+3 d^2 i\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (30 e^2 i+49 c^2 \left (e^2 g+3 d e h+3 d^2 i\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (e h+3 d i) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b e^3 i x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b \left (32 \left (11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+720 e^3 i+1176 c^2 e \left (e^2 g+3 d e h+3 d^2 i\right )\right )+3675 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{352800 c^7}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) \text {ArcSin}(c x)}{96 c^6}+d^3 f x (a+b \text {ArcSin}(c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \text {ArcSin}(c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \text {ArcSin}(c x))+\frac {1}{4} \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right ) x^4 (a+b \text {ArcSin}(c x))+\frac {1}{5} e \left (e^2 g+3 d e h+3 d^2 i\right ) x^5 (a+b \text {ArcSin}(c x))+\frac {1}{6} e^2 (e h+3 d i) x^6 (a+b \text {ArcSin}(c x))+\frac {1}{7} e^3 i x^7 (a+b \text {ArcSin}(c x)) \]
[Out]
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Rubi [A]
time = 3.17, antiderivative size = 684, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4833, 12, 1823,
794, 222} \begin {gather*} d^3 f x (a+b \text {ArcSin}(c x))+\frac {1}{3} d x^3 (a+b \text {ArcSin}(c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \text {ArcSin}(c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \text {ArcSin}(c x))+\frac {1}{4} x^4 (a+b \text {ArcSin}(c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \text {ArcSin}(c x))+\frac {1}{7} e^3 i x^7 (a+b \text {ArcSin}(c x))-\frac {b \text {ArcSin}(c x) \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{96 c^6}+\frac {b e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{36 c}+\frac {b e^3 i x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{1225 c^3}+\frac {b x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{144 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )}{11025 c^5}+\frac {b \sqrt {1-c^2 x^2} \left (3675 c^2 x \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )+32 \left (11025 c^6 d^3 f+2450 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+1176 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+720 e^3 i\right )\right )}{352800 c^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 222
Rule 794
Rule 1823
Rule 4833
Rubi steps
\begin {align*} \int (d+e x)^3 \left (f+g x+h x^2+106 x^3\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (70 d^3 (6 f+x (3 g+x (2 h+159 x)))+21 d e^2 x^2 (20 f+x (15 g+4 x (3 h+265 x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+424 x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+636 x)))\right )}{420 \sqrt {1-c^2 x^2}} \, dx\\ &=d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{420} (b c) \int \frac {x \left (70 d^3 (6 f+x (3 g+x (2 h+159 x)))+21 d e^2 x^2 (20 f+x (15 g+4 x (3 h+265 x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+424 x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+636 x)))\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-2940 c^2 d^3 f-1470 c^2 d^2 (3 e f+d g) x-980 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right ) x^2-735 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^3-12 e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4-490 c^2 e^2 (318 d+e h) x^5\right )}{\sqrt {1-c^2 x^2}} \, dx}{2940 c}\\ &=\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (17640 c^4 d^3 f+8820 c^4 d^2 (3 e f+d g) x+5880 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right ) x^2+490 c^2 \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3+72 c^2 e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{17640 c^3}\\ &=\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-88200 c^6 d^3 f-44100 c^6 d^2 (3 e f+d g) x-24 c^2 \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2-2450 c^4 \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3\right )}{\sqrt {1-c^2 x^2}} \, dx}{88200 c^5}\\ &=\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (352800 c^8 d^3 f+7350 c^4 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x+96 c^4 \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{352800 c^7}\\ &=\frac {b \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-96 c^4 \left (76320 e^3+11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+1176 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right )-22050 c^6 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{1058400 c^9}\\ &=\frac {b \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b \left (32 \left (76320 e^3+11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+1176 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right )+3675 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{352800 c^7}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac {b \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b \left (32 \left (76320 e^3+11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+1176 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right )+3675 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{352800 c^7}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) \sin ^{-1}(c x)}{96 c^6}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 619, normalized size = 0.90 \begin {gather*} a d^3 f x+\frac {1}{2} a d^2 (3 e f+d g) x^2+\frac {1}{3} a d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3+\frac {1}{4} a \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right ) x^4+\frac {1}{5} a e \left (e^2 g+3 d e h+3 d^2 i\right ) x^5+\frac {1}{6} a e^2 (e h+3 d i) x^6+\frac {1}{7} a e^3 i x^7+\frac {b \sqrt {1-c^2 x^2} \left (23040 e^3 i+3 c^2 e \left (37632 d^2 i+147 d e (256 h+125 i x)+e^2 (12544 g+5 x (1225 h+768 i x))\right )+c^4 \left (1225 d^3 (64 h+27 i x)+147 d^2 e \left (1600 g+675 h x+384 i x^2\right )+147 d e^2 \left (1600 f+x \left (675 g+384 h x+250 i x^2\right )\right )+e^3 x \left (33075 f+2 x \left (9408 g+6125 h x+4320 i x^2\right )\right )\right )+2 c^6 \left (1225 d^3 (144 f+x (36 g+x (16 h+9 i x)))+147 d^2 e x (900 f+x (400 g+9 x (25 h+16 i x)))+147 d e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))+e^3 x^3 (11025 f+4 x (1764 g+25 x (49 h+36 i x)))\right )\right )}{352800 c^7}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) \text {ArcSin}(c x)}{96 c^6}+\frac {1}{420} b x \left (35 d^3 (12 f+x (6 g+x (4 h+3 i x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+4 i x)))+21 d e^2 x^2 (20 f+x (15 g+2 x (6 h+5 i x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+6 i x)))\right ) \text {ArcSin}(c x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 0.15, size = 932, normalized size = 1.36 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
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Maxima [A]
time = 0.49, size = 1215, normalized size = 1.78 \begin {gather*} \frac {1}{6} \, a h x^{6} e^{3} + \frac {1}{7} i \, a x^{7} e^{3} + \frac {3}{5} \, a d h x^{5} e^{2} + \frac {1}{2} i \, a d x^{6} e^{2} + \frac {3}{4} \, a d^{2} h x^{4} e + \frac {3}{5} i \, a d^{2} x^{5} e + \frac {1}{3} \, a d^{3} h x^{3} + \frac {1}{4} i \, a d^{3} x^{4} + \frac {1}{5} \, a g x^{5} e^{3} + \frac {3}{4} \, a d g x^{4} e^{2} + a d^{2} g x^{3} e + \frac {1}{2} \, a d^{3} g x^{2} + \frac {1}{4} \, a f x^{4} e^{3} + a d f x^{3} e^{2} + \frac {3}{2} \, a d^{2} f x^{2} e + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} h + a d^{3} f x + \frac {3}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} f e + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} g e + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} h e + \frac {1}{32} i \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{3} + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3} f}{c} + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d f e^{2} + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d g e^{2} + \frac {1}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d h e^{2} + \frac {1}{25} i \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} e + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b f e^{3} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b g e^{3} + \frac {1}{288} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b h e^{3} + \frac {1}{96} i \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b d e^{2} + \frac {1}{245} i \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.54, size = 885, normalized size = 1.29 \begin {gather*} \frac {235200 \, a c^{7} d^{3} h x^{3} + 176400 i \, a c^{7} d^{3} x^{4} + 352800 \, a c^{7} d^{3} g x^{2} + 705600 \, a c^{7} d^{3} f x + 1680 \, {\left (70 \, a c^{7} h x^{6} + 60 i \, a c^{7} x^{7} + 84 \, a c^{7} g x^{5} + 105 \, a c^{7} f x^{4}\right )} e^{3} + 35280 \, {\left (12 \, a c^{7} d h x^{5} + 10 i \, a c^{7} d x^{6} + 15 \, a c^{7} d g x^{4} + 20 \, a c^{7} d f x^{3}\right )} e^{2} + 35280 \, {\left (15 \, a c^{7} d^{2} h x^{4} + 12 i \, a c^{7} d^{2} x^{5} + 20 \, a c^{7} d^{2} g x^{3} + 30 \, a c^{7} d^{2} f x^{2}\right )} e - 105 \, {\left (1120 i \, b c^{7} d^{3} h x^{3} - 840 \, b c^{7} d^{3} x^{4} + 1680 i \, b c^{7} d^{3} g x^{2} + 3360 i \, b c^{7} d^{3} f x - 840 i \, b c^{5} d^{3} g + 315 \, b c^{3} d^{3} + {\left (560 i \, b c^{7} h x^{6} - 480 \, b c^{7} x^{7} + 672 i \, b c^{7} g x^{5} + 840 i \, b c^{7} f x^{4} - 315 i \, b c^{3} f - 175 i \, b c h\right )} e^{3} + 21 \, {\left (96 i \, b c^{7} d h x^{5} - 80 \, b c^{7} d x^{6} + 120 i \, b c^{7} d g x^{4} + 160 i \, b c^{7} d f x^{3} - 45 i \, b c^{3} d g + 25 \, b c d\right )} e^{2} + 21 \, {\left (120 i \, b c^{7} d^{2} h x^{4} - 96 \, b c^{7} d^{2} x^{5} + 160 i \, b c^{7} d^{2} g x^{3} + 240 i \, b c^{7} d^{2} f x^{2} - 120 i \, b c^{5} d^{2} f - 45 i \, b c^{3} d^{2} h\right )} e\right )} \log \left (-2 \, c^{2} x^{2} - 2 \, \sqrt {c^{2} x^{2} - 1} c x + 1\right ) - 2 \, {\left (-39200 i \, b c^{6} d^{3} h x^{2} + 22050 \, b c^{6} d^{3} x^{3} - 352800 i \, b c^{6} d^{3} f - 78400 i \, b c^{4} d^{3} h + 11025 \, {\left (-8 i \, b c^{6} d^{3} g + 3 \, b c^{4} d^{3}\right )} x + {\left (-9800 i \, b c^{6} h x^{5} + 7200 \, b c^{6} x^{6} + 288 \, {\left (-49 i \, b c^{6} g + 30 \, b c^{4}\right )} x^{4} - 37632 i \, b c^{2} g + 2450 \, {\left (-9 i \, b c^{6} f - 5 i \, b c^{4} h\right )} x^{3} + 384 \, {\left (-49 i \, b c^{4} g + 30 \, b c^{2}\right )} x^{2} + 3675 \, {\left (-9 i \, b c^{4} f - 5 i \, b c^{2} h\right )} x + 23040 \, b\right )} e^{3} + 147 \, {\left (-288 i \, b c^{6} d h x^{4} + 200 \, b c^{6} d x^{5} - 1600 i \, b c^{4} d f - 768 i \, b c^{2} d h + 50 \, {\left (-9 i \, b c^{6} d g + 5 \, b c^{4} d\right )} x^{3} + 32 \, {\left (-25 i \, b c^{6} d f - 12 i \, b c^{4} d h\right )} x^{2} + 75 \, {\left (-9 i \, b c^{4} d g + 5 \, b c^{2} d\right )} x\right )} e^{2} + 147 \, {\left (-450 i \, b c^{6} d^{2} h x^{3} + 288 \, b c^{6} d^{2} x^{4} - 1600 i \, b c^{4} d^{2} g + 768 \, b c^{2} d^{2} + 32 \, {\left (-25 i \, b c^{6} d^{2} g + 12 \, b c^{4} d^{2}\right )} x^{2} + 225 \, {\left (-8 i \, b c^{6} d^{2} f - 3 i \, b c^{4} d^{2} h\right )} x\right )} e\right )} \sqrt {c^{2} x^{2} - 1}}{705600 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1809 vs.
\(2 (688) = 1376\).
time = 1.19, size = 1809, normalized size = 2.64 \begin {gather*} \begin {cases} a d^{3} f x + \frac {a d^{3} g x^{2}}{2} + \frac {a d^{3} h x^{3}}{3} + \frac {a d^{3} i x^{4}}{4} + \frac {3 a d^{2} e f x^{2}}{2} + a d^{2} e g x^{3} + \frac {3 a d^{2} e h x^{4}}{4} + \frac {3 a d^{2} e i x^{5}}{5} + a d e^{2} f x^{3} + \frac {3 a d e^{2} g x^{4}}{4} + \frac {3 a d e^{2} h x^{5}}{5} + \frac {a d e^{2} i x^{6}}{2} + \frac {a e^{3} f x^{4}}{4} + \frac {a e^{3} g x^{5}}{5} + \frac {a e^{3} h x^{6}}{6} + \frac {a e^{3} i x^{7}}{7} + b d^{3} f x \operatorname {asin}{\left (c x \right )} + \frac {b d^{3} g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b d^{3} h x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d^{3} i x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {3 b d^{2} e f x^{2} \operatorname {asin}{\left (c x \right )}}{2} + b d^{2} e g x^{3} \operatorname {asin}{\left (c x \right )} + \frac {3 b d^{2} e h x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {3 b d^{2} e i x^{5} \operatorname {asin}{\left (c x \right )}}{5} + b d e^{2} f x^{3} \operatorname {asin}{\left (c x \right )} + \frac {3 b d e^{2} g x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {3 b d e^{2} h x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b d e^{2} i x^{6} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e^{3} f x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b e^{3} g x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b e^{3} h x^{6} \operatorname {asin}{\left (c x \right )}}{6} + \frac {b e^{3} i x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {b d^{3} f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d^{3} g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d^{3} h x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b d^{3} i x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {3 b d^{2} e f x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d^{2} e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {3 b d^{2} e h x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {3 b d^{2} e i x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b d e^{2} f x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {3 b d e^{2} g x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {3 b d e^{2} h x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b d e^{2} i x^{5} \sqrt {- c^{2} x^{2} + 1}}{12 c} + \frac {b e^{3} f x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b e^{3} g x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b e^{3} h x^{5} \sqrt {- c^{2} x^{2} + 1}}{36 c} + \frac {b e^{3} i x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} - \frac {b d^{3} g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {3 b d^{2} e f \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {2 b d^{3} h \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b d^{3} i x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {2 b d^{2} e g \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + \frac {9 b d^{2} e h x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b d^{2} e i x^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{3}} + \frac {2 b d e^{2} f \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + \frac {9 b d e^{2} g x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b d e^{2} h x^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{3}} + \frac {5 b d e^{2} i x^{3} \sqrt {- c^{2} x^{2} + 1}}{48 c^{3}} + \frac {3 b e^{3} f x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b e^{3} g x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {5 b e^{3} h x^{3} \sqrt {- c^{2} x^{2} + 1}}{144 c^{3}} + \frac {6 b e^{3} i x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} - \frac {3 b d^{3} i \operatorname {asin}{\left (c x \right )}}{32 c^{4}} - \frac {9 b d^{2} e h \operatorname {asin}{\left (c x \right )}}{32 c^{4}} - \frac {9 b d e^{2} g \operatorname {asin}{\left (c x \right )}}{32 c^{4}} - \frac {3 b e^{3} f \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {8 b d^{2} e i \sqrt {- c^{2} x^{2} + 1}}{25 c^{5}} + \frac {8 b d e^{2} h \sqrt {- c^{2} x^{2} + 1}}{25 c^{5}} + \frac {5 b d e^{2} i x \sqrt {- c^{2} x^{2} + 1}}{32 c^{5}} + \frac {8 b e^{3} g \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + \frac {5 b e^{3} h x \sqrt {- c^{2} x^{2} + 1}}{96 c^{5}} + \frac {8 b e^{3} i x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} - \frac {5 b d e^{2} i \operatorname {asin}{\left (c x \right )}}{32 c^{6}} - \frac {5 b e^{3} h \operatorname {asin}{\left (c x \right )}}{96 c^{6}} + \frac {16 b e^{3} i \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\a \left (d^{3} f x + \frac {d^{3} g x^{2}}{2} + \frac {d^{3} h x^{3}}{3} + \frac {d^{3} i x^{4}}{4} + \frac {3 d^{2} e f x^{2}}{2} + d^{2} e g x^{3} + \frac {3 d^{2} e h x^{4}}{4} + \frac {3 d^{2} e i x^{5}}{5} + d e^{2} f x^{3} + \frac {3 d e^{2} g x^{4}}{4} + \frac {3 d e^{2} h x^{5}}{5} + \frac {d e^{2} i x^{6}}{2} + \frac {e^{3} f x^{4}}{4} + \frac {e^{3} g x^{5}}{5} + \frac {e^{3} h x^{6}}{6} + \frac {e^{3} i x^{7}}{7}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2010 vs.
\(2 (643) = 1286\).
time = 0.44, size = 2010, normalized size = 2.94 \begin {gather*} \text {Too large to display} \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 (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3\,\left (i\,x^3+h\,x^2+g\,x+f\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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