3.2016 \(\int \frac{(a+b x) (d+e x)^3}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{(a+b x) (d+e x)^4}{4 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

((a + b*x)*(d + e*x)^4)/(4*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.128484, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(a+b x) (d+e x)^4}{4 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(d + e*x)^3)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((a + b*x)*(d + e*x)^4)/(4*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 18.1598, size = 34, normalized size = 0.87 \[ \frac{\left (a + b x\right ) \left (d + e x\right )^{4}}{4 e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**3/((b*x+a)**2)**(1/2),x)

[Out]

(a + b*x)*(d + e*x)**4/(4*e*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0223387, size = 30, normalized size = 0.77 \[ \frac{(a+b x) (d+e x)^4}{4 e \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(d + e*x)^3)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((a + b*x)*(d + e*x)^4)/(4*e*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.005, size = 47, normalized size = 1.2 \[{\frac{x \left ({e}^{3}{x}^{3}+4\,{e}^{2}{x}^{2}d+6\,{d}^{2}ex+4\,{d}^{3} \right ) \left ( bx+a \right ) }{4}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^3/((b*x+a)^2)^(1/2),x)

[Out]

1/4*x*(e^3*x^3+4*d*e^2*x^2+6*d^2*e*x+4*d^3)*(b*x+a)/((b*x+a)^2)^(1/2)

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Maxima [A]  time = 0.696924, size = 713, normalized size = 18.28 \[ \frac{13 \, a^{4} b e^{3} \log \left (x + \frac{a}{b}\right )}{6 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{13 \, a^{3} e^{3} x}{6 \,{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{13 \, a^{2} e^{3} x^{2}}{12 \, \sqrt{b^{2}} b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e^{3} x^{3}}{4 \, b} + a \sqrt{\frac{1}{b^{2}}} d^{3} \log \left (x + \frac{a}{b}\right ) - \frac{7 \, a^{4} \sqrt{\frac{1}{b^{2}}} e^{3} \log \left (x + \frac{a}{b}\right )}{6 \, b^{3}} - \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{3} x^{2}}{12 \, b^{2}} - \frac{5 \,{\left (3 \, b d e^{2} + a e^{3}\right )} a^{3} b \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{3 \,{\left (b d^{2} e + a d e^{2}\right )} a^{2} b^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{3}}{6 \, b^{4}} + \frac{5 \,{\left (3 \, b d e^{2} + a e^{3}\right )} a^{2} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{3 \,{\left (b d^{2} e + a d e^{2}\right )} a b x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (b d^{2} e + a d e^{2}\right )} x^{2}}{2 \, \sqrt{b^{2}}} - \frac{5 \,{\left (3 \, b d e^{2} + a e^{3}\right )} a x^{2}}{6 \, \sqrt{b^{2}} b} + \frac{2 \,{\left (3 \, b d e^{2} + a e^{3}\right )} a^{3} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} - \frac{{\left (b d^{3} + 3 \, a d^{2} e\right )} a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} + \frac{{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} - \frac{2 \,{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} + \frac{{\left (b d^{3} + 3 \, a d^{2} e\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^3/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

13/6*a^4*b*e^3*log(x + a/b)/(b^2)^(5/2) - 13/6*a^3*e^3*x/(b^2)^(3/2) + 13/12*a^2
*e^3*x^2/(sqrt(b^2)*b) + 1/4*sqrt(b^2*x^2 + 2*a*b*x + a^2)*e^3*x^3/b + a*sqrt(b^
(-2))*d^3*log(x + a/b) - 7/6*a^4*sqrt(b^(-2))*e^3*log(x + a/b)/b^3 - 7/12*sqrt(b
^2*x^2 + 2*a*b*x + a^2)*a*e^3*x^2/b^2 - 5/3*(3*b*d*e^2 + a*e^3)*a^3*b*log(x + a/
b)/(b^2)^(5/2) + 3*(b*d^2*e + a*d*e^2)*a^2*b^2*log(x + a/b)/(b^2)^(5/2) + 7/6*sq
rt(b^2*x^2 + 2*a*b*x + a^2)*a^3*e^3/b^4 + 5/3*(3*b*d*e^2 + a*e^3)*a^2*x/(b^2)^(3
/2) - 3*(b*d^2*e + a*d*e^2)*a*b*x/(b^2)^(3/2) + 3/2*(b*d^2*e + a*d*e^2)*x^2/sqrt
(b^2) - 5/6*(3*b*d*e^2 + a*e^3)*a*x^2/(sqrt(b^2)*b) + 2/3*(3*b*d*e^2 + a*e^3)*a^
3*sqrt(b^(-2))*log(x + a/b)/b^3 - (b*d^3 + 3*a*d^2*e)*a*sqrt(b^(-2))*log(x + a/b
)/b + 1/3*(3*b*d*e^2 + a*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*x^2/b^2 - 2/3*(3*b*d
*e^2 + a*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2/b^4 + (b*d^3 + 3*a*d^2*e)*sqrt(b
^2*x^2 + 2*a*b*x + a^2)/b^2

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Fricas [A]  time = 0.271566, size = 42, normalized size = 1.08 \[ \frac{1}{4} \, e^{3} x^{4} + d e^{2} x^{3} + \frac{3}{2} \, d^{2} e x^{2} + d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^3/sqrt((b*x + a)^2),x, algorithm="fricas")

[Out]

1/4*e^3*x^4 + d*e^2*x^3 + 3/2*d^2*e*x^2 + d^3*x

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Sympy [A]  time = 0.209568, size = 32, normalized size = 0.82 \[ d^{3} x + \frac{3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac{e^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**3/((b*x+a)**2)**(1/2),x)

[Out]

d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4

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GIAC/XCAS [A]  time = 0.277648, size = 24, normalized size = 0.62 \[ \frac{1}{4} \,{\left (x e + d\right )}^{4} e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^3/sqrt((b*x + a)^2),x, algorithm="giac")

[Out]

1/4*(x*e + d)^4*e^(-1)*sign(b*x + a)