Optimal. Leaf size=350 \[ -\frac{2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^8 d^4 (p+1)}+\frac{2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^8 d^4 (p+2)}-\frac{6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^8 d^4 (p+3)}-\frac{10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^8 d^4 (p+5)}+\frac{6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^8 d^4 (p+6)}+\frac{2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^8 d^4 (p+4)}-\frac{14 a \left (a+b \sqrt{c+d x}\right )^{p+7}}{b^8 d^4 (p+7)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+8}}{b^8 d^4 (p+8)} \]
[Out]
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Rubi [A] time = 0.616953, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^8 d^4 (p+1)}+\frac{2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^8 d^4 (p+2)}-\frac{6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^8 d^4 (p+3)}-\frac{10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^8 d^4 (p+5)}+\frac{6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^8 d^4 (p+6)}+\frac{2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^8 d^4 (p+4)}-\frac{14 a \left (a+b \sqrt{c+d x}\right )^{p+7}}{b^8 d^4 (p+7)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+8}}{b^8 d^4 (p+8)} \]
Antiderivative was successfully verified.
[In] Int[x^3*(a + b*Sqrt[c + d*x])^p,x]
[Out]
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Rubi in Sympy [A] time = 49.9932, size = 320, normalized size = 0.91 \[ - \frac{2 a \left (a + b \sqrt{c + d x}\right )^{p + 1} \left (a^{2} - b^{2} c\right )^{3}}{b^{8} d^{4} \left (p + 1\right )} - \frac{6 a \left (a + b \sqrt{c + d x}\right )^{p + 3} \left (a^{2} - b^{2} c\right ) \left (7 a^{2} - 3 b^{2} c\right )}{b^{8} d^{4} \left (p + 3\right )} - \frac{10 a \left (a + b \sqrt{c + d x}\right )^{p + 5} \left (7 a^{2} - 3 b^{2} c\right )}{b^{8} d^{4} \left (p + 5\right )} - \frac{14 a \left (a + b \sqrt{c + d x}\right )^{p + 7}}{b^{8} d^{4} \left (p + 7\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 2} \left (a^{2} - b^{2} c\right )^{2} \left (7 a^{2} - b^{2} c\right )}{b^{8} d^{4} \left (p + 2\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 4} \left (35 a^{4} - 30 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{b^{8} d^{4} \left (p + 4\right )} + \frac{6 \left (a + b \sqrt{c + d x}\right )^{p + 6} \left (7 a^{2} - b^{2} c\right )}{b^{8} d^{4} \left (p + 6\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 8}}{b^{8} d^{4} \left (p + 8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(a+b*(d*x+c)**(1/2))**p,x)
[Out]
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Mathematica [A] time = 1.48302, size = 554, normalized size = 1.58 \[ -\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+1} \left (5040 a^7-5040 a^6 b (p+1) \sqrt{c+d x}-360 a^5 b^2 \left (-6 c \left (p^2+p-7\right )-7 d \left (p^2+3 p+2\right ) x\right )+120 a^4 b^3 (p+1) \sqrt{c+d x} \left (c \left (-4 p^2+10 p+126\right )-7 d \left (p^2+5 p+6\right ) x\right )+6 a^3 b^4 \left (8 c^2 \left (p^4-14 p^3-139 p^2-124 p+315\right )+40 c d \left (p^4+4 p^3-16 p^2-61 p-42\right ) x+35 d^2 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^2\right )-6 a^2 b^5 (p+1) \sqrt{c+d x} \left (-24 c^2 \left (p^3+5 p^2-24 p-105\right )+4 c d \left (p^4-p^3-94 p^2-386 p-420\right ) x+7 d^2 \left (p^4+14 p^3+71 p^2+154 p+120\right ) x^2\right )+a b^6 \left (48 c^3 \left (3 p^4+38 p^3+138 p^2+103 p-105\right )-24 c^2 d \left (2 p^5+24 p^4+74 p^3-21 p^2-283 p-210\right ) x+6 c d^2 \left (p^6+11 p^5+10 p^4-265 p^3-1151 p^2-1726 p-840\right ) x^2+7 d^3 \left (p^6+21 p^5+175 p^4+735 p^3+1624 p^2+1764 p+720\right ) x^3\right )+b^7 \left (p^4+16 p^3+86 p^2+176 p+105\right ) \sqrt{c+d x} \left (48 c^3-24 c^2 d (p+2) x+6 c d^2 \left (p^2+6 p+8\right ) x^2-d^3 \left (p^3+12 p^2+44 p+48\right ) x^3\right )\right )}{b^8 d^4 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6) (p+7) (p+8)} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(a + b*Sqrt[c + d*x])^p,x]
[Out]
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Maple [F] time = 0.008, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( a+b\sqrt{dx+c} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(a+b*(d*x+c)^(1/2))^p,x)
[Out]
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Maxima [A] time = 0.723975, size = 983, normalized size = 2.81 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^p*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.439962, size = 1912, normalized size = 5.46 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^p*x^3,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(a+b*(d*x+c)**(1/2))**p,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^p*x^3,x, algorithm="giac")
[Out]