3.1971 \(\int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=219 \[ \frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{2 b^5}+\frac{6 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^5}+\frac{2 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{3 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^4}{5 b^5}+\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5} \]

[Out]

((b*d - a*e)^4*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*b^5) + (2*e*(b*d -
a*e)^3*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^5) + (6*e^2*(b*d - a*e)^2
*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^5) + (e^3*(b*d - a*e)*(a + b*x)
^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^5) + (e^4*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(9*b^5)

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Rubi [A]  time = 0.51466, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{2 b^5}+\frac{6 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^5}+\frac{2 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{3 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^4}{5 b^5}+\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((b*d - a*e)^4*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*b^5) + (2*e*(b*d -
a*e)^3*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^5) + (6*e^2*(b*d - a*e)^2
*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^5) + (e^3*(b*d - a*e)*(a + b*x)
^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^5) + (e^4*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(9*b^5)

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Rubi in Sympy [A]  time = 62.5219, size = 182, normalized size = 0.83 \[ \frac{\left (d + e x\right )^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{9 b} - \frac{\left (d + e x\right )^{3} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{18 b^{2}} + \frac{\left (d + e x\right )^{2} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{42 b^{3}} - \frac{\left (d + e x\right ) \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{126 b^{4}} + \frac{\left (a e - b d\right )^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{630 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

(d + e*x)**4*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(9*b) - (d + e*x)**3*(a*e - b*d
)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(18*b**2) + (d + e*x)**2*(a*e - b*d)**2*(a
**2 + 2*a*b*x + b**2*x**2)**(5/2)/(42*b**3) - (d + e*x)*(a*e - b*d)**3*(a**2 + 2
*a*b*x + b**2*x**2)**(5/2)/(126*b**4) + (a*e - b*d)**4*(a**2 + 2*a*b*x + b**2*x*
*2)**(5/2)/(630*b**5)

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Mathematica [A]  time = 0.171309, size = 267, normalized size = 1.22 \[ \frac{x \sqrt{(a+b x)^2} \left (126 a^4 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+84 a^3 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+36 a^2 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+9 a b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )\right )}{630 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(126*a^4*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3
+ e^4*x^4) + 84*a^3*b*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 +
5*e^4*x^4) + 36*a^2*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x
^3 + 15*e^4*x^4) + 9*a*b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e
^3*x^3 + 35*e^4*x^4) + b^4*x^4*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*
e^3*x^3 + 70*e^4*x^4)))/(630*(a + b*x))

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Maple [B]  time = 0.01, size = 339, normalized size = 1.6 \[{\frac{x \left ( 70\,{b}^{4}{e}^{4}{x}^{8}+315\,{x}^{7}a{b}^{3}{e}^{4}+315\,{x}^{7}{b}^{4}d{e}^{3}+540\,{x}^{6}{a}^{2}{b}^{2}{e}^{4}+1440\,{x}^{6}a{b}^{3}d{e}^{3}+540\,{x}^{6}{b}^{4}{d}^{2}{e}^{2}+420\,{x}^{5}{a}^{3}b{e}^{4}+2520\,{x}^{5}{a}^{2}{b}^{2}d{e}^{3}+2520\,{x}^{5}a{b}^{3}{d}^{2}{e}^{2}+420\,{x}^{5}{b}^{4}{d}^{3}e+126\,{x}^{4}{a}^{4}{e}^{4}+2016\,{x}^{4}{a}^{3}bd{e}^{3}+4536\,{x}^{4}{a}^{2}{b}^{2}{d}^{2}{e}^{2}+2016\,{x}^{4}a{b}^{3}{d}^{3}e+126\,{x}^{4}{b}^{4}{d}^{4}+630\,{a}^{4}d{e}^{3}{x}^{3}+3780\,{a}^{3}b{d}^{2}{e}^{2}{x}^{3}+3780\,{a}^{2}{b}^{2}{d}^{3}e{x}^{3}+630\,a{b}^{3}{d}^{4}{x}^{3}+1260\,{x}^{2}{a}^{4}{d}^{2}{e}^{2}+3360\,{x}^{2}{a}^{3}b{d}^{3}e+1260\,{x}^{2}{b}^{2}{d}^{4}{a}^{2}+1260\,{a}^{4}{d}^{3}ex+1260\,b{d}^{4}{a}^{3}x+630\,{a}^{4}{d}^{4} \right ) }{630\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/630*x*(70*b^4*e^4*x^8+315*a*b^3*e^4*x^7+315*b^4*d*e^3*x^7+540*a^2*b^2*e^4*x^6+
1440*a*b^3*d*e^3*x^6+540*b^4*d^2*e^2*x^6+420*a^3*b*e^4*x^5+2520*a^2*b^2*d*e^3*x^
5+2520*a*b^3*d^2*e^2*x^5+420*b^4*d^3*e*x^5+126*a^4*e^4*x^4+2016*a^3*b*d*e^3*x^4+
4536*a^2*b^2*d^2*e^2*x^4+2016*a*b^3*d^3*e*x^4+126*b^4*d^4*x^4+630*a^4*d*e^3*x^3+
3780*a^3*b*d^2*e^2*x^3+3780*a^2*b^2*d^3*e*x^3+630*a*b^3*d^4*x^3+1260*a^4*d^2*e^2
*x^2+3360*a^3*b*d^3*e*x^2+1260*a^2*b^2*d^4*x^2+1260*a^4*d^3*e*x+1260*a^3*b*d^4*x
+630*a^4*d^4)*((b*x+a)^2)^(3/2)/(b*x+a)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)*(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.27279, size = 385, normalized size = 1.76 \[ \frac{1}{9} \, b^{4} e^{4} x^{9} + a^{4} d^{4} x + \frac{1}{2} \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{8} + \frac{2}{7} \,{\left (3 \, b^{4} d^{2} e^{2} + 8 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{7} + \frac{2}{3} \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{4} + 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} + 16 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} x^{5} +{\left (a b^{3} d^{4} + 6 \, a^{2} b^{2} d^{3} e + 6 \, a^{3} b d^{2} e^{2} + a^{4} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} d^{4} + 8 \, a^{3} b d^{3} e + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} + 2 \,{\left (a^{3} b d^{4} + a^{4} d^{3} e\right )} x^{2} \]

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)*(e*x + d)^4,x, algorithm="fricas")

[Out]

1/9*b^4*e^4*x^9 + a^4*d^4*x + 1/2*(b^4*d*e^3 + a*b^3*e^4)*x^8 + 2/7*(3*b^4*d^2*e
^2 + 8*a*b^3*d*e^3 + 3*a^2*b^2*e^4)*x^7 + 2/3*(b^4*d^3*e + 6*a*b^3*d^2*e^2 + 6*a
^2*b^2*d*e^3 + a^3*b*e^4)*x^6 + 1/5*(b^4*d^4 + 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e
^2 + 16*a^3*b*d*e^3 + a^4*e^4)*x^5 + (a*b^3*d^4 + 6*a^2*b^2*d^3*e + 6*a^3*b*d^2*
e^2 + a^4*d*e^3)*x^4 + 2/3*(3*a^2*b^2*d^4 + 8*a^3*b*d^3*e + 3*a^4*d^2*e^2)*x^3 +
2*(a^3*b*d^4 + a^4*d^3*e)*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right ) \left (d + e x\right )^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**4*((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.290713, size = 622, normalized size = 2.84 \[ \frac{1}{9} \, b^{4} x^{9} e^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, b^{4} d x^{8} e^{3}{\rm sign}\left (b x + a\right ) + \frac{6}{7} \, b^{4} d^{2} x^{7} e^{2}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, b^{4} d^{3} x^{6} e{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, b^{4} d^{4} x^{5}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a b^{3} x^{8} e^{4}{\rm sign}\left (b x + a\right ) + \frac{16}{7} \, a b^{3} d x^{7} e^{3}{\rm sign}\left (b x + a\right ) + 4 \, a b^{3} d^{2} x^{6} e^{2}{\rm sign}\left (b x + a\right ) + \frac{16}{5} \, a b^{3} d^{3} x^{5} e{\rm sign}\left (b x + a\right ) + a b^{3} d^{4} x^{4}{\rm sign}\left (b x + a\right ) + \frac{6}{7} \, a^{2} b^{2} x^{7} e^{4}{\rm sign}\left (b x + a\right ) + 4 \, a^{2} b^{2} d x^{6} e^{3}{\rm sign}\left (b x + a\right ) + \frac{36}{5} \, a^{2} b^{2} d^{2} x^{5} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{3} x^{4} e{\rm sign}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{4} x^{3}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, a^{3} b x^{6} e^{4}{\rm sign}\left (b x + a\right ) + \frac{16}{5} \, a^{3} b d x^{5} e^{3}{\rm sign}\left (b x + a\right ) + 6 \, a^{3} b d^{2} x^{4} e^{2}{\rm sign}\left (b x + a\right ) + \frac{16}{3} \, a^{3} b d^{3} x^{3} e{\rm sign}\left (b x + a\right ) + 2 \, a^{3} b d^{4} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, a^{4} x^{5} e^{4}{\rm sign}\left (b x + a\right ) + a^{4} d x^{4} e^{3}{\rm sign}\left (b x + a\right ) + 2 \, a^{4} d^{2} x^{3} e^{2}{\rm sign}\left (b x + a\right ) + 2 \, a^{4} d^{3} x^{2} e{\rm sign}\left (b x + a\right ) + a^{4} d^{4} x{\rm sign}\left (b x + a\right ) \]

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)*(e*x + d)^4,x, algorithm="giac")

[Out]

1/9*b^4*x^9*e^4*sign(b*x + a) + 1/2*b^4*d*x^8*e^3*sign(b*x + a) + 6/7*b^4*d^2*x^
7*e^2*sign(b*x + a) + 2/3*b^4*d^3*x^6*e*sign(b*x + a) + 1/5*b^4*d^4*x^5*sign(b*x
+ a) + 1/2*a*b^3*x^8*e^4*sign(b*x + a) + 16/7*a*b^3*d*x^7*e^3*sign(b*x + a) + 4
*a*b^3*d^2*x^6*e^2*sign(b*x + a) + 16/5*a*b^3*d^3*x^5*e*sign(b*x + a) + a*b^3*d^
4*x^4*sign(b*x + a) + 6/7*a^2*b^2*x^7*e^4*sign(b*x + a) + 4*a^2*b^2*d*x^6*e^3*si
gn(b*x + a) + 36/5*a^2*b^2*d^2*x^5*e^2*sign(b*x + a) + 6*a^2*b^2*d^3*x^4*e*sign(
b*x + a) + 2*a^2*b^2*d^4*x^3*sign(b*x + a) + 2/3*a^3*b*x^6*e^4*sign(b*x + a) + 1
6/5*a^3*b*d*x^5*e^3*sign(b*x + a) + 6*a^3*b*d^2*x^4*e^2*sign(b*x + a) + 16/3*a^3
*b*d^3*x^3*e*sign(b*x + a) + 2*a^3*b*d^4*x^2*sign(b*x + a) + 1/5*a^4*x^5*e^4*sig
n(b*x + a) + a^4*d*x^4*e^3*sign(b*x + a) + 2*a^4*d^2*x^3*e^2*sign(b*x + a) + 2*a
^4*d^3*x^2*e*sign(b*x + a) + a^4*d^4*x*sign(b*x + a)