3.808 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=316 \[ -\frac{20 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt{x} (a+b x)}+\frac{2 b^4 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{3 (a+b x)}+\frac{10 a b^3 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac{2 b^5 B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}-\frac{2 a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{5 x^{5/2} (a+b x)}-\frac{10 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 x^{3/2} (a+b x)} \]

[Out]

(-2*a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^(7/2)*(a + b*x)) - (2*a^4*(5*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^(5/2)*(a + b*x)) - (10*a^3*b*(2*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x)) - (20*a^2*b^2*(A*b + a
*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(Sqrt[x]*(a + b*x)) + (10*a*b^3*(A*b + 2*a*B)
*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*b^4*(A*b + 5*a*B)*x^(3/2)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*b^5*B*x^(5/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(5*(a + b*x))

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Rubi [A]  time = 0.337432, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{20 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt{x} (a+b x)}+\frac{2 b^4 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{3 (a+b x)}+\frac{10 a b^3 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac{2 b^5 B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}-\frac{2 a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{5 x^{5/2} (a+b x)}-\frac{10 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^(9/2),x]

[Out]

(-2*a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^(7/2)*(a + b*x)) - (2*a^4*(5*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^(5/2)*(a + b*x)) - (10*a^3*b*(2*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x)) - (20*a^2*b^2*(A*b + a
*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(Sqrt[x]*(a + b*x)) + (10*a*b^3*(A*b + 2*a*B)
*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*b^4*(A*b + 5*a*B)*x^(3/2)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*b^5*B*x^(5/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(5*(a + b*x))

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Rubi in Sympy [A]  time = 34.4728, size = 313, normalized size = 0.99 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{7 a x^{\frac{7}{2}}} + \frac{512 a b^{3} \sqrt{x} \left (5 A b + 7 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105 \left (a + b x\right )} + \frac{256 b^{3} \sqrt{x} \left (5 A b + 7 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105} + \frac{64 b^{3} \sqrt{x} \left (a + b x\right ) \left (5 A b + 7 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 a} - \frac{32 b^{2} \left (5 A b + 7 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{21 a \sqrt{x}} - \frac{4 b \left (a + b x\right ) \left (5 A b + 7 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{21 a x^{\frac{3}{2}}} - \frac{2 \left (5 A b + 7 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{35 a x^{\frac{5}{2}}} \]

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)**(5/2)/x**(9/2),x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(7*a*x**(7/2)) + 512*a*b**3
*sqrt(x)*(5*A*b + 7*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(105*(a + b*x)) + 256*
b**3*sqrt(x)*(5*A*b + 7*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/105 + 64*b**3*sqrt
(x)*(a + b*x)*(5*A*b + 7*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*a) - 32*b**2*
(5*A*b + 7*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(21*a*sqrt(x)) - 4*b*(a + b*
x)*(5*A*b + 7*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(21*a*x**(3/2)) - 2*(5*A*
b + 7*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(35*a*x**(5/2))

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Mathematica [A]  time = 0.0887201, size = 122, normalized size = 0.39 \[ -\frac{2 \sqrt{(a+b x)^2} \left (3 a^5 (5 A+7 B x)+35 a^4 b x (3 A+5 B x)+350 a^3 b^2 x^2 (A+3 B x)+1050 a^2 b^3 x^3 (A-B x)-175 a b^4 x^4 (3 A+B x)-7 b^5 x^5 (5 A+3 B x)\right )}{105 x^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^(9/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(1050*a^2*b^3*x^3*(A - B*x) - 175*a*b^4*x^4*(3*A + B*x) +
350*a^3*b^2*x^2*(A + 3*B*x) - 7*b^5*x^5*(5*A + 3*B*x) + 35*a^4*b*x*(3*A + 5*B*x)
+ 3*a^5*(5*A + 7*B*x)))/(105*x^(7/2)*(a + b*x))

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Maple [A]  time = 0.011, size = 140, normalized size = 0.4 \[ -{\frac{-42\,B{b}^{5}{x}^{6}-70\,A{x}^{5}{b}^{5}-350\,B{x}^{5}a{b}^{4}-1050\,A{x}^{4}a{b}^{4}-2100\,B{x}^{4}{a}^{2}{b}^{3}+2100\,A{x}^{3}{a}^{2}{b}^{3}+2100\,B{x}^{3}{a}^{3}{b}^{2}+700\,A{x}^{2}{a}^{3}{b}^{2}+350\,B{x}^{2}{a}^{4}b+210\,Ax{a}^{4}b+42\,Bx{a}^{5}+30\,A{a}^{5}}{105\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{7}{2}}}} \]

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)^(5/2)/x^(9/2),x)

[Out]

-2/105*(-21*B*b^5*x^6-35*A*b^5*x^5-175*B*a*b^4*x^5-525*A*a*b^4*x^4-1050*B*a^2*b^
3*x^4+1050*A*a^2*b^3*x^3+1050*B*a^3*b^2*x^3+350*A*a^3*b^2*x^2+175*B*a^4*b*x^2+10
5*A*a^4*b*x+21*B*a^5*x+15*A*a^5)*((b*x+a)^2)^(5/2)/x^(7/2)/(b*x+a)^5

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Maxima [A]  time = 0.712427, size = 316, normalized size = 1. \[ \frac{2}{15} \,{\left ({\left (3 \, b^{5} x^{2} + 5 \, a b^{4} x\right )} \sqrt{x} + \frac{20 \,{\left (a b^{4} x^{2} + 3 \, a^{2} b^{3} x\right )}}{\sqrt{x}} + \frac{90 \,{\left (a^{2} b^{3} x^{2} - a^{3} b^{2} x\right )}}{x^{\frac{3}{2}}} - \frac{20 \,{\left (3 \, a^{3} b^{2} x^{2} + a^{4} b x\right )}}{x^{\frac{5}{2}}} - \frac{5 \, a^{4} b x^{2} + 3 \, a^{5} x}{x^{\frac{7}{2}}}\right )} B + \frac{2}{105} \, A{\left (\frac{35 \,{\left (b^{5} x^{2} + 3 \, a b^{4} x\right )}}{\sqrt{x}} + \frac{420 \,{\left (a b^{4} x^{2} - a^{2} b^{3} x\right )}}{x^{\frac{3}{2}}} - \frac{210 \,{\left (3 \, a^{2} b^{3} x^{2} + a^{3} b^{2} x\right )}}{x^{\frac{5}{2}}} - \frac{28 \,{\left (5 \, a^{3} b^{2} x^{2} + 3 \, a^{4} b x\right )}}{x^{\frac{7}{2}}} - \frac{3 \,{\left (7 \, a^{4} b x^{2} + 5 \, a^{5} x\right )}}{x^{\frac{9}{2}}}\right )} \]

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^(9/2),x, algorithm="maxima")

[Out]

2/15*((3*b^5*x^2 + 5*a*b^4*x)*sqrt(x) + 20*(a*b^4*x^2 + 3*a^2*b^3*x)/sqrt(x) + 9
0*(a^2*b^3*x^2 - a^3*b^2*x)/x^(3/2) - 20*(3*a^3*b^2*x^2 + a^4*b*x)/x^(5/2) - (5*
a^4*b*x^2 + 3*a^5*x)/x^(7/2))*B + 2/105*A*(35*(b^5*x^2 + 3*a*b^4*x)/sqrt(x) + 42
0*(a*b^4*x^2 - a^2*b^3*x)/x^(3/2) - 210*(3*a^2*b^3*x^2 + a^3*b^2*x)/x^(5/2) - 28
*(5*a^3*b^2*x^2 + 3*a^4*b*x)/x^(7/2) - 3*(7*a^4*b*x^2 + 5*a^5*x)/x^(9/2))

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Fricas [A]  time = 0.273165, size = 161, normalized size = 0.51 \[ \frac{2 \,{\left (21 \, B b^{5} x^{6} - 15 \, A a^{5} + 35 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 525 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 1050 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 175 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 21 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )}}{105 \, x^{\frac{7}{2}}} \]

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^(9/2),x, algorithm="fricas")

[Out]

2/105*(21*B*b^5*x^6 - 15*A*a^5 + 35*(5*B*a*b^4 + A*b^5)*x^5 + 525*(2*B*a^2*b^3 +
A*a*b^4)*x^4 - 1050*(B*a^3*b^2 + A*a^2*b^3)*x^3 - 175*(B*a^4*b + 2*A*a^3*b^2)*x
^2 - 21*(B*a^5 + 5*A*a^4*b)*x)/x^(7/2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(5/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278701, size = 265, normalized size = 0.84 \[ \frac{2}{5} \, B b^{5} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, B a b^{4} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, A b^{5} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 20 \, B a^{2} b^{3} \sqrt{x}{\rm sign}\left (b x + a\right ) + 10 \, A a b^{4} \sqrt{x}{\rm sign}\left (b x + a\right ) - \frac{2 \,{\left (1050 \, B a^{3} b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 1050 \, A a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 175 \, B a^{4} b x^{2}{\rm sign}\left (b x + a\right ) + 350 \, A a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 21 \, B a^{5} x{\rm sign}\left (b x + a\right ) + 105 \, A a^{4} b x{\rm sign}\left (b x + a\right ) + 15 \, A a^{5}{\rm sign}\left (b x + a\right )\right )}}{105 \, x^{\frac{7}{2}}} \]

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^(9/2),x, algorithm="giac")

[Out]

2/5*B*b^5*x^(5/2)*sign(b*x + a) + 10/3*B*a*b^4*x^(3/2)*sign(b*x + a) + 2/3*A*b^5
*x^(3/2)*sign(b*x + a) + 20*B*a^2*b^3*sqrt(x)*sign(b*x + a) + 10*A*a*b^4*sqrt(x)
*sign(b*x + a) - 2/105*(1050*B*a^3*b^2*x^3*sign(b*x + a) + 1050*A*a^2*b^3*x^3*si
gn(b*x + a) + 175*B*a^4*b*x^2*sign(b*x + a) + 350*A*a^3*b^2*x^2*sign(b*x + a) +
21*B*a^5*x*sign(b*x + a) + 105*A*a^4*b*x*sign(b*x + a) + 15*A*a^5*sign(b*x + a))
/x^(7/2)