Optimal. Leaf size=512 \[ \frac{1}{4} e x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac{1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac{1}{2} d^2 x^2 (d g+3 e f) \left (a+b \sin ^{-1}(c x)\right )+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 (3 d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b e x^3 \sqrt{1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{144 c^3}+\frac{b x^2 \sqrt{1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )}{225 c^3}+\frac{b \sqrt{1-c^2 x^2} \left (75 x \left (9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+24 c^4 d^2 (d g+3 e f)+5 e^3 h\right )+32 \left (50 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+225 c^4 d^3 f+24 e^2 (3 d h+e g)\right )\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+24 c^4 d^2 (d g+3 e f)+5 e^3 h\right )}{96 c^6}+\frac{b e^2 x^4 \sqrt{1-c^2 x^2} (3 d h+e g)}{25 c}+\frac{b e^3 h x^5 \sqrt{1-c^2 x^2}}{36 c} \]
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Rubi [A] time = 2.51078, antiderivative size = 509, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4749, 12, 1809, 780, 216} \[ \frac{1}{4} e x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac{1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac{1}{2} d^2 x^2 (d g+3 e f) \left (a+b \sin ^{-1}(c x)\right )+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 (3 d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b e x^3 \sqrt{1-c^2 x^2} \left (e^2 \left (\frac{5 h}{c^2}+9 f\right )+27 d^2 h+27 d e g\right )}{144 c}+\frac{b x^2 \sqrt{1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )}{225 c^3}+\frac{b \sqrt{1-c^2 x^2} \left (75 x \left (9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+24 c^4 d^2 (d g+3 e f)+5 e^3 h\right )+32 \left (50 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+225 c^4 d^3 f+24 e^2 (3 d h+e g)\right )\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+24 c^4 d^2 (d g+3 e f)+5 e^3 h\right )}{96 c^6}+\frac{b e^2 x^4 \sqrt{1-c^2 x^2} (3 d h+e g)}{25 c}+\frac{b e^3 h x^5 \sqrt{1-c^2 x^2}}{36 c} \]
Antiderivative was successfully verified.
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Rule 4749
Rule 12
Rule 1809
Rule 780
Rule 216
Rubi steps
\begin{align*} \int (d+e x)^3 \left (f+g x+h x^2\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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x \left (10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right )}{60 \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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{60} (b c) \int \frac{x \left (10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b e^3 h x^5 \sqrt{1-c^2 x^2}}{36 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-360 c^2 d^3 f-180 c^2 d^2 (3 e f+d g) x-120 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right ) x^2-10 e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3-72 c^2 e^2 (e g+3 d h) x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{360 c}\\ &=\frac{b e^2 (e g+3 d h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b e^3 h x^5 \sqrt{1-c^2 x^2}}{36 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \int \frac{x \left (1800 c^4 d^3 f+900 c^4 d^2 (3 e f+d g) x+24 c^2 \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2+50 c^2 e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3\right )}{\sqrt{1-c^2 x^2}} \, dx}{1800 c^3}\\ &=\frac{b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e^2 (e g+3 d h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b e^3 h x^5 \sqrt{1-c^2 x^2}}{36 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-7200 c^6 d^3 f-150 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x-96 c^4 \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{7200 c^5}\\ &=\frac{b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e^2 (e g+3 d h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b e^3 h x^5 \sqrt{1-c^2 x^2}}{36 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \int \frac{x \left (96 c^4 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+450 c^4 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right )}{\sqrt{1-c^2 x^2}} \, dx}{21600 c^7}\\ &=\frac{b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e^2 (e g+3 d h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b e^3 h x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{b \left (32 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+75 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right ) \sqrt{1-c^2 x^2}}{7200 c^5}+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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac{b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e^2 (e g+3 d h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b e^3 h x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{b \left (32 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+75 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right ) \sqrt{1-c^2 x^2}}{7200 c^5}-\frac{b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.518424, size = 463, normalized size = 0.9 \[ \frac{1}{4} a e x^4 \left (3 d^2 h+3 d e g+e^2 f\right )+\frac{1}{3} a d x^3 \left (d^2 h+3 d e g+3 e^2 f\right )+\frac{1}{2} a d^2 x^2 (d g+3 e f)+a d^3 f x+\frac{1}{5} a e^2 x^5 (3 d h+e g)+\frac{1}{6} a e^3 h x^6+\frac{b \sqrt{1-c^2 x^2} \left (2 c^4 \left (75 d^2 e x (36 f+x (16 g+9 h x))+100 d^3 (36 f+x (9 g+4 h x))+3 d e^2 x^2 (400 f+9 x (25 g+16 h x))+e^3 x^3 (225 f+4 x (36 g+25 h x))\right )+c^2 \left (75 d^2 e (64 g+27 h x)+1600 d^3 h+3 d e^2 \left (1600 f+675 g x+384 h x^2\right )+e^3 x \left (675 f+384 g x+250 h x^2\right )\right )+3 e^2 (768 d h+256 e g+125 e h x)\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+24 c^4 d^2 (d g+3 e f)+5 e^3 h\right )}{96 c^6}+\frac{1}{60} b x \sin ^{-1}(c x) \left (15 d^2 e x (6 f+x (4 g+3 h x))+10 d^3 (6 f+x (3 g+2 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 705, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68132, size = 1257, normalized size = 2.46 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.03276, size = 1509, normalized size = 2.95 \begin{align*} \frac{1200 \, a c^{6} e^{3} h x^{6} + 7200 \, a c^{6} d^{3} f x + 1440 \,{\left (a c^{6} e^{3} g + 3 \, a c^{6} d e^{2} h\right )} x^{5} + 1800 \,{\left (a c^{6} e^{3} f + 3 \, a c^{6} d e^{2} g + 3 \, a c^{6} d^{2} e h\right )} x^{4} + 2400 \,{\left (3 \, a c^{6} d e^{2} f + 3 \, a c^{6} d^{2} e g + a c^{6} d^{3} h\right )} x^{3} + 3600 \,{\left (3 \, a c^{6} d^{2} e f + a c^{6} d^{3} g\right )} x^{2} + 15 \,{\left (80 \, b c^{6} e^{3} h x^{6} + 480 \, b c^{6} d^{3} f x + 96 \,{\left (b c^{6} e^{3} g + 3 \, b c^{6} d e^{2} h\right )} x^{5} + 120 \,{\left (b c^{6} e^{3} f + 3 \, b c^{6} d e^{2} g + 3 \, b c^{6} d^{2} e h\right )} x^{4} + 160 \,{\left (3 \, b c^{6} d e^{2} f + 3 \, b c^{6} d^{2} e g + b c^{6} d^{3} h\right )} x^{3} + 240 \,{\left (3 \, b c^{6} d^{2} e f + b c^{6} d^{3} g\right )} x^{2} - 45 \,{\left (8 \, b c^{4} d^{2} e + b c^{2} e^{3}\right )} f - 15 \,{\left (8 \, b c^{4} d^{3} + 9 \, b c^{2} d e^{2}\right )} g - 5 \,{\left (27 \, b c^{2} d^{2} e + 5 \, b e^{3}\right )} h\right )} \arcsin \left (c x\right ) +{\left (200 \, b c^{5} e^{3} h x^{5} + 288 \,{\left (b c^{5} e^{3} g + 3 \, b c^{5} d e^{2} h\right )} x^{4} + 50 \,{\left (9 \, b c^{5} e^{3} f + 27 \, b c^{5} d e^{2} g +{\left (27 \, b c^{5} d^{2} e + 5 \, b c^{3} e^{3}\right )} h\right )} x^{3} + 32 \,{\left (75 \, b c^{5} d e^{2} f + 3 \,{\left (25 \, b c^{5} d^{2} e + 4 \, b c^{3} e^{3}\right )} g +{\left (25 \, b c^{5} d^{3} + 36 \, b c^{3} d e^{2}\right )} h\right )} x^{2} + 2400 \,{\left (3 \, b c^{5} d^{3} + 2 \, b c^{3} d e^{2}\right )} f + 192 \,{\left (25 \, b c^{3} d^{2} e + 4 \, b c e^{3}\right )} g + 64 \,{\left (25 \, b c^{3} d^{3} + 36 \, b c d e^{2}\right )} h + 75 \,{\left (9 \,{\left (8 \, b c^{5} d^{2} e + b c^{3} e^{3}\right )} f + 3 \,{\left (8 \, b c^{5} d^{3} + 9 \, b c^{3} d e^{2}\right )} g +{\left (27 \, b c^{3} d^{2} e + 5 \, b c e^{3}\right )} h\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{7200 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.0425, size = 1263, normalized size = 2.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32452, size = 1962, normalized size = 3.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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