3.1576 \(\int \frac{(d+e x)^4}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{(a+b x) (d+e x)^3 (b d-a e)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^4}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^4 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x (a+b x) (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2 (b d-a e)^2}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(e*(b*d - a*e)^3*x*(a + b*x))/(b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((b*d - a*e)
^2*(a + b*x)*(d + e*x)^2)/(2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((b*d - a*e)*(
a + b*x)*(d + e*x)^3)/(3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((a + b*x)*(d + e*
x)^4)/(4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((b*d - a*e)^4*(a + b*x)*Log[a + b*x
])/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.229694, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{(a+b x) (d+e x)^3 (b d-a e)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^4}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^4 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x (a+b x) (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2 (b d-a e)^2}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(e*(b*d - a*e)^3*x*(a + b*x))/(b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((b*d - a*e)
^2*(a + b*x)*(d + e*x)^2)/(2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((b*d - a*e)*(
a + b*x)*(d + e*x)^3)/(3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((a + b*x)*(d + e*
x)^4)/(4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((b*d - a*e)^4*(a + b*x)*Log[a + b*x
])/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 45.4087, size = 212, normalized size = 0.95 \[ \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{4}}{8 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{3} \left (a e - b d\right )}{6 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{2} \left (a e - b d\right )^{2}}{4 b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{e \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{5}} + \frac{\left (a + b x\right ) \left (a e - b d\right )^{4} \log{\left (a + b x \right )}}{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/((b*x+a)**2)**(1/2),x)

[Out]

(2*a + 2*b*x)*(d + e*x)**4/(8*b*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - (2*a + 2*b*x
)*(d + e*x)**3*(a*e - b*d)/(6*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + (2*a + 2*
b*x)*(d + e*x)**2*(a*e - b*d)**2/(4*b**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - e*(
a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/b**5 + (a + b*x)*(a*e - b*d)**4*l
og(a + b*x)/(b**5*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.145327, size = 130, normalized size = 0.59 \[ \frac{(a+b x) \left (b e x \left (-12 a^3 e^3+6 a^2 b e^2 (8 d+e x)-4 a b^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+b^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (a+b x)\right )}{12 b^5 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*(b*e*x*(-12*a^3*e^3 + 6*a^2*b*e^2*(8*d + e*x) - 4*a*b^2*e*(18*d^2 + 6
*d*e*x + e^2*x^2) + b^3*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3)) + 12*(
b*d - a*e)^4*Log[a + b*x]))/(12*b^5*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.013, size = 223, normalized size = 1. \[{\frac{ \left ( bx+a \right ) \left ( 3\,{x}^{4}{b}^{4}{e}^{4}-4\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}+6\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-24\,{x}^{2}a{b}^{3}d{e}^{3}+36\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( bx+a \right ){a}^{4}{e}^{4}-48\,\ln \left ( bx+a \right ){a}^{3}bd{e}^{3}+72\,\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-48\,\ln \left ( bx+a \right ) a{b}^{3}{d}^{3}e+12\,\ln \left ( bx+a \right ){b}^{4}{d}^{4}-12\,x{a}^{3}b{e}^{4}+48\,x{a}^{2}{b}^{2}d{e}^{3}-72\,xa{b}^{3}{d}^{2}{e}^{2}+48\,x{b}^{4}{d}^{3}e \right ) }{12\,{b}^{5}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/((b*x+a)^2)^(1/2),x)

[Out]

1/12*(b*x+a)*(3*x^4*b^4*e^4-4*x^3*a*b^3*e^4+16*x^3*b^4*d*e^3+6*x^2*a^2*b^2*e^4-2
4*x^2*a*b^3*d*e^3+36*x^2*b^4*d^2*e^2+12*ln(b*x+a)*a^4*e^4-48*ln(b*x+a)*a^3*b*d*e
^3+72*ln(b*x+a)*a^2*b^2*d^2*e^2-48*ln(b*x+a)*a*b^3*d^3*e+12*ln(b*x+a)*b^4*d^4-12
*x*a^3*b*e^4+48*x*a^2*b^2*d*e^3-72*x*a*b^3*d^2*e^2+48*x*b^4*d^3*e)/((b*x+a)^2)^(
1/2)/b^5

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Maxima [A]  time = 0.695328, size = 590, normalized size = 2.66 \[ \frac{6 \, a^{2} b^{2} d^{2} e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{20 \, a^{3} b d e^{3} \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{13 \, a^{4} e^{4} \log \left (x + \frac{a}{b}\right )}{6 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{6 \, a b d^{2} e^{2} x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{20 \, a^{2} d e^{3} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{13 \, a^{3} e^{4} x}{6 \,{\left (b^{2}\right )}^{\frac{3}{2}} b} + \frac{3 \, d^{2} e^{2} x^{2}}{\sqrt{b^{2}}} - \frac{10 \, a d e^{3} x^{2}}{3 \, \sqrt{b^{2}} b} + \frac{13 \, a^{2} e^{4} x^{2}}{12 \, \sqrt{b^{2}} b^{2}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e^{4} x^{3}}{4 \, b^{2}} + \sqrt{\frac{1}{b^{2}}} d^{4} \log \left (x + \frac{a}{b}\right ) - \frac{4 \, a \sqrt{\frac{1}{b^{2}}} d^{3} e \log \left (x + \frac{a}{b}\right )}{b} + \frac{8 \, a^{3} \sqrt{\frac{1}{b^{2}}} d e^{3} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} - \frac{7 \, a^{4} \sqrt{\frac{1}{b^{2}}} e^{4} \log \left (x + \frac{a}{b}\right )}{6 \, b^{4}} + \frac{4 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} d e^{3} x^{2}}{3 \, b^{2}} - \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{4} x^{2}}{12 \, b^{3}} + \frac{4 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} d^{3} e}{b^{2}} - \frac{8 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d e^{3}}{3 \, b^{4}} + \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{4}}{6 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

6*a^2*b^2*d^2*e^2*log(x + a/b)/(b^2)^(5/2) - 20/3*a^3*b*d*e^3*log(x + a/b)/(b^2)
^(5/2) + 13/6*a^4*e^4*log(x + a/b)/(b^2)^(5/2) - 6*a*b*d^2*e^2*x/(b^2)^(3/2) + 2
0/3*a^2*d*e^3*x/(b^2)^(3/2) - 13/6*a^3*e^4*x/((b^2)^(3/2)*b) + 3*d^2*e^2*x^2/sqr
t(b^2) - 10/3*a*d*e^3*x^2/(sqrt(b^2)*b) + 13/12*a^2*e^4*x^2/(sqrt(b^2)*b^2) + 1/
4*sqrt(b^2*x^2 + 2*a*b*x + a^2)*e^4*x^3/b^2 + sqrt(b^(-2))*d^4*log(x + a/b) - 4*
a*sqrt(b^(-2))*d^3*e*log(x + a/b)/b + 8/3*a^3*sqrt(b^(-2))*d*e^3*log(x + a/b)/b^
3 - 7/6*a^4*sqrt(b^(-2))*e^4*log(x + a/b)/b^4 + 4/3*sqrt(b^2*x^2 + 2*a*b*x + a^2
)*d*e^3*x^2/b^2 - 7/12*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*e^4*x^2/b^3 + 4*sqrt(b^2*
x^2 + 2*a*b*x + a^2)*d^3*e/b^2 - 8/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*d*e^3/b^4
+ 7/6*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3*e^4/b^5

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Fricas [A]  time = 0.205852, size = 244, normalized size = 1.1 \[ \frac{3 \, b^{4} e^{4} x^{4} + 4 \,{\left (4 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (6 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 12 \,{\left (4 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 4 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/sqrt((b*x + a)^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.22585, size = 134, normalized size = 0.6 \[ \frac{e^{4} x^{4}}{4 b} - \frac{x^{3} \left (a e^{4} - 4 b d e^{3}\right )}{3 b^{2}} + \frac{x^{2} \left (a^{2} e^{4} - 4 a b d e^{3} + 6 b^{2} d^{2} e^{2}\right )}{2 b^{3}} - \frac{x \left (a^{3} e^{4} - 4 a^{2} b d e^{3} + 6 a b^{2} d^{2} e^{2} - 4 b^{3} d^{3} e\right )}{b^{4}} + \frac{\left (a e - b d\right )^{4} \log{\left (a + b x \right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/((b*x+a)**2)**(1/2),x)

[Out]

e**4*x**4/(4*b) - x**3*(a*e**4 - 4*b*d*e**3)/(3*b**2) + x**2*(a**2*e**4 - 4*a*b*
d*e**3 + 6*b**2*d**2*e**2)/(2*b**3) - x*(a**3*e**4 - 4*a**2*b*d*e**3 + 6*a*b**2*
d**2*e**2 - 4*b**3*d**3*e)/b**4 + (a*e - b*d)**4*log(a + b*x)/b**5

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GIAC/XCAS [A]  time = 0.215257, size = 356, normalized size = 1.6 \[ \frac{3 \, b^{3} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 16 \, b^{3} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 36 \, b^{3} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 48 \, b^{3} d^{3} x e{\rm sign}\left (b x + a\right ) - 4 \, a b^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) - 24 \, a b^{2} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) - 72 \, a b^{2} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 48 \, a^{2} b d x e^{3}{\rm sign}\left (b x + a\right ) - 12 \, a^{3} x e^{4}{\rm sign}\left (b x + a\right )}{12 \, b^{4}} + \frac{{\left (b^{4} d^{4}{\rm sign}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (b x + a\right ) + a^{4} e^{4}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/sqrt((b*x + a)^2),x, algorithm="giac")

[Out]

1/12*(3*b^3*x^4*e^4*sign(b*x + a) + 16*b^3*d*x^3*e^3*sign(b*x + a) + 36*b^3*d^2*
x^2*e^2*sign(b*x + a) + 48*b^3*d^3*x*e*sign(b*x + a) - 4*a*b^2*x^3*e^4*sign(b*x
+ a) - 24*a*b^2*d*x^2*e^3*sign(b*x + a) - 72*a*b^2*d^2*x*e^2*sign(b*x + a) + 6*a
^2*b*x^2*e^4*sign(b*x + a) + 48*a^2*b*d*x*e^3*sign(b*x + a) - 12*a^3*x*e^4*sign(
b*x + a))/b^4 + (b^4*d^4*sign(b*x + a) - 4*a*b^3*d^3*e*sign(b*x + a) + 6*a^2*b^2
*d^2*e^2*sign(b*x + a) - 4*a^3*b*d*e^3*sign(b*x + a) + a^4*e^4*sign(b*x + a))*ln
(abs(b*x + a))/b^5