Optimal. Leaf size=264 \[ \frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{5 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^5 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^5 (a+b x)} \]
[Out]
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Rubi [A] time = 0.300887, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{5 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^5 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^5 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 39.2506, size = 226, normalized size = 0.86 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{11 e} + \frac{16 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99 e^{2}} + \frac{32 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 e^{3}} + \frac{128 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{1155 e^{4}} + \frac{256 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3465 e^{5} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)*(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.161532, size = 172, normalized size = 0.65 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (1155 a^4 e^4+924 a^3 b e^3 (3 e x-2 d)+198 a^2 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+44 a b^3 e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.012, size = 202, normalized size = 0.8 \[{\frac{630\,{x}^{4}{b}^{4}{e}^{4}+3080\,{x}^{3}a{b}^{3}{e}^{4}-560\,{x}^{3}{b}^{4}d{e}^{3}+5940\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-2640\,{x}^{2}a{b}^{3}d{e}^{3}+480\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+5544\,x{a}^{3}b{e}^{4}-4752\,x{a}^{2}{b}^{2}d{e}^{3}+2112\,xa{b}^{3}{d}^{2}{e}^{2}-384\,x{b}^{4}{d}^{3}e+2310\,{a}^{4}{e}^{4}-3696\,{a}^{3}bd{e}^{3}+3168\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-1408\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{3465\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)*(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.721307, size = 518, normalized size = 1.96 \[ \frac{2 \,{\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \,{\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} +{\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt{e x + d} a}{315 \, e^{4}} + \frac{2 \,{\left (315 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 528 \, a b^{2} d^{4} e + 792 \, a^{2} b d^{3} e^{2} - 462 \, a^{3} d^{2} e^{3} + 35 \,{\left (b^{3} d e^{4} + 33 \, a b^{2} e^{5}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{2} e^{3} - 33 \, a b^{2} d e^{4} - 297 \, a^{2} b e^{5}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{3} e^{2} - 66 \, a b^{2} d^{2} e^{3} + 99 \, a^{2} b d e^{4} + 231 \, a^{3} e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{4} e - 264 \, a b^{2} d^{3} e^{2} + 396 \, a^{2} b d^{2} e^{3} - 231 \, a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d} b}{3465 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.28468, size = 331, normalized size = 1.25 \[ \frac{2 \,{\left (315 \, b^{4} e^{5} x^{5} + 128 \, b^{4} d^{5} - 704 \, a b^{3} d^{4} e + 1584 \, a^{2} b^{2} d^{3} e^{2} - 1848 \, a^{3} b d^{2} e^{3} + 1155 \, a^{4} d e^{4} + 35 \,{\left (b^{4} d e^{4} + 44 \, a b^{3} e^{5}\right )} x^{4} - 10 \,{\left (4 \, b^{4} d^{2} e^{3} - 22 \, a b^{3} d e^{4} - 297 \, a^{2} b^{2} e^{5}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{3} e^{2} - 44 \, a b^{3} d^{2} e^{3} + 99 \, a^{2} b^{2} d e^{4} + 462 \, a^{3} b e^{5}\right )} x^{2} -{\left (64 \, b^{4} d^{4} e - 352 \, a b^{3} d^{3} e^{2} + 792 \, a^{2} b^{2} d^{2} e^{3} - 924 \, a^{3} b d e^{4} - 1155 \, a^{4} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)*sqrt(e*x + d),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right ) \sqrt{d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(3/2)*(e*x+d)**(1/2),x)
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GIAC/XCAS [A] time = 0.333885, size = 365, normalized size = 1.38 \[ \frac{2}{3465} \,{\left (924 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{3} b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 198 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a^{2} b^{2} e^{\left (-14\right )}{\rm sign}\left (b x + a\right ) + 44 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} a b^{3} e^{\left (-27\right )}{\rm sign}\left (b x + a\right ) +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} b^{4} e^{\left (-44\right )}{\rm sign}\left (b x + a\right ) + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)*sqrt(e*x + d),x, algorithm="giac")
[Out]