3.685 \(\int x^4 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=306 \[ \frac{5 a^2 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 (a+b x)}+\frac{b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{10 (a+b x)}+\frac{5 a b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac{b^5 B x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{a^5 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a^4 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{6 (a+b x)}+\frac{5 a^3 b x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 (a+b x)} \]

[Out]

(a^5*A*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a^4*(5*A*b + a*B)*x^6
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(a + b*x)) + (5*a^3*b*(2*A*b + a*B)*x^7*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (5*a^2*b^2*(A*b + a*B)*x^8*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (5*a*b^3*(A*b + 2*a*B)*x^9*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(9*(a + b*x)) + (b^4*(A*b + 5*a*B)*x^10*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(10*(a + b*x)) + (b^5*B*x^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x))

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Rubi [A]  time = 0.444995, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{5 a^2 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 (a+b x)}+\frac{b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{10 (a+b x)}+\frac{5 a b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac{b^5 B x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{a^5 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a^4 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{6 (a+b x)}+\frac{5 a^3 b x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(a^5*A*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a^4*(5*A*b + a*B)*x^6
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(a + b*x)) + (5*a^3*b*(2*A*b + a*B)*x^7*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (5*a^2*b^2*(A*b + a*B)*x^8*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (5*a*b^3*(A*b + 2*a*B)*x^9*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(9*(a + b*x)) + (b^4*(A*b + 5*a*B)*x^10*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(10*(a + b*x)) + (b^5*B*x^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x))

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Rubi in Sympy [A]  time = 39.1005, size = 270, normalized size = 0.88 \[ \frac{B x^{5} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{22 b} + \frac{a^{4} \left (2 a + 2 b x\right ) \left (11 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{3960 b^{6}} - \frac{a^{3} \left (11 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{2310 b^{6}} + \frac{a^{2} x^{2} \left (2 a + 2 b x\right ) \left (11 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{1320 b^{4}} - \frac{a x^{3} \left (2 a + 2 b x\right ) \left (11 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{495 b^{3}} + \frac{x^{4} \left (2 a + 2 b x\right ) \left (11 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{220 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

B*x**5*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(22*b) + a**4*(2*a + 2*
b*x)*(11*A*b - 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(3960*b**6) - a**3*(11
*A*b - 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(2310*b**6) + a**2*x**2*(2*a +
2*b*x)*(11*A*b - 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(1320*b**4) - a*x**
3*(2*a + 2*b*x)*(11*A*b - 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(495*b**3)
+ x**4*(2*a + 2*b*x)*(11*A*b - 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(220*b
**2)

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Mathematica [A]  time = 0.0837517, size = 125, normalized size = 0.41 \[ \frac{x^5 \sqrt{(a+b x)^2} \left (462 a^5 (6 A+5 B x)+1650 a^4 b x (7 A+6 B x)+2475 a^3 b^2 x^2 (8 A+7 B x)+1925 a^2 b^3 x^3 (9 A+8 B x)+770 a b^4 x^4 (10 A+9 B x)+126 b^5 x^5 (11 A+10 B x)\right )}{13860 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x^5*Sqrt[(a + b*x)^2]*(462*a^5*(6*A + 5*B*x) + 1650*a^4*b*x*(7*A + 6*B*x) + 247
5*a^3*b^2*x^2*(8*A + 7*B*x) + 1925*a^2*b^3*x^3*(9*A + 8*B*x) + 770*a*b^4*x^4*(10
*A + 9*B*x) + 126*b^5*x^5*(11*A + 10*B*x)))/(13860*(a + b*x))

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Maple [A]  time = 0.01, size = 140, normalized size = 0.5 \[{\frac{{x}^{5} \left ( 1260\,B{b}^{5}{x}^{6}+1386\,{x}^{5}A{b}^{5}+6930\,{x}^{5}Ba{b}^{4}+7700\,{x}^{4}Aa{b}^{4}+15400\,{x}^{4}B{a}^{2}{b}^{3}+17325\,{x}^{3}A{a}^{2}{b}^{3}+17325\,{x}^{3}B{a}^{3}{b}^{2}+19800\,{x}^{2}A{a}^{3}{b}^{2}+9900\,{x}^{2}B{a}^{4}b+11550\,xA{a}^{4}b+2310\,xB{a}^{5}+2772\,A{a}^{5} \right ) }{13860\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/13860*x^5*(1260*B*b^5*x^6+1386*A*b^5*x^5+6930*B*a*b^4*x^5+7700*A*a*b^4*x^4+154
00*B*a^2*b^3*x^4+17325*A*a^2*b^3*x^3+17325*B*a^3*b^2*x^3+19800*A*a^3*b^2*x^2+990
0*B*a^4*b*x^2+11550*A*a^4*b*x+2310*B*a^5*x+2772*A*a^5)*((b*x+a)^2)^(5/2)/(b*x+a)
^5

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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)^(5/2)*(B*x + A)*x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283239, size = 161, normalized size = 0.53 \[ \frac{1}{11} \, B b^{5} x^{11} + \frac{1}{5} \, A a^{5} x^{5} + \frac{1}{10} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac{5}{4} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} + \frac{5}{7} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^4,x, algorithm="fricas")

[Out]

1/11*B*b^5*x^11 + 1/5*A*a^5*x^5 + 1/10*(5*B*a*b^4 + A*b^5)*x^10 + 5/9*(2*B*a^2*b
^3 + A*a*b^4)*x^9 + 5/4*(B*a^3*b^2 + A*a^2*b^3)*x^8 + 5/7*(B*a^4*b + 2*A*a^3*b^2
)*x^7 + 1/6*(B*a^5 + 5*A*a^4*b)*x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x**4*(A + B*x)*((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.273232, size = 300, normalized size = 0.98 \[ \frac{1}{11} \, B b^{5} x^{11}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, B a b^{4} x^{10}{\rm sign}\left (b x + a\right ) + \frac{1}{10} \, A b^{5} x^{10}{\rm sign}\left (b x + a\right ) + \frac{10}{9} \, B a^{2} b^{3} x^{9}{\rm sign}\left (b x + a\right ) + \frac{5}{9} \, A a b^{4} x^{9}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, B a^{3} b^{2} x^{8}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, A a^{2} b^{3} x^{8}{\rm sign}\left (b x + a\right ) + \frac{5}{7} \, B a^{4} b x^{7}{\rm sign}\left (b x + a\right ) + \frac{10}{7} \, A a^{3} b^{2} x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{6} \, B a^{5} x^{6}{\rm sign}\left (b x + a\right ) + \frac{5}{6} \, A a^{4} b x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, A a^{5} x^{5}{\rm sign}\left (b x + a\right ) - \frac{{\left (5 \, B a^{11} - 11 \, A a^{10} b\right )}{\rm sign}\left (b x + a\right )}{13860 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^4,x, algorithm="giac")

[Out]

1/11*B*b^5*x^11*sign(b*x + a) + 1/2*B*a*b^4*x^10*sign(b*x + a) + 1/10*A*b^5*x^10
*sign(b*x + a) + 10/9*B*a^2*b^3*x^9*sign(b*x + a) + 5/9*A*a*b^4*x^9*sign(b*x + a
) + 5/4*B*a^3*b^2*x^8*sign(b*x + a) + 5/4*A*a^2*b^3*x^8*sign(b*x + a) + 5/7*B*a^
4*b*x^7*sign(b*x + a) + 10/7*A*a^3*b^2*x^7*sign(b*x + a) + 1/6*B*a^5*x^6*sign(b*
x + a) + 5/6*A*a^4*b*x^6*sign(b*x + a) + 1/5*A*a^5*x^5*sign(b*x + a) - 1/13860*(
5*B*a^11 - 11*A*a^10*b)*sign(b*x + a)/b^6