3.798 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{7/2}} \, dx\)

Optimal. Leaf size=216 \[ -\frac{6 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt{x} (a+b x)}+\frac{2 b^2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{a+b x}-\frac{2 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{3 x^{3/2} (a+b x)}+\frac{2 b^3 B x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)} \]

[Out]

(-2*a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^(5/2)*(a + b*x)) - (2*a^2*(3*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x)) - (6*a*b*(A*b + a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(Sqrt[x]*(a + b*x)) + (2*b^2*(A*b + 3*a*B)*Sqrt[x
]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*b^3*B*x^(3/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(3*(a + b*x))

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Rubi [A]  time = 0.24966, antiderivative size = 216, 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 \[ -\frac{6 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt{x} (a+b x)}+\frac{2 b^2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{a+b x}-\frac{2 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{3 x^{3/2} (a+b x)}+\frac{2 b^3 B x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^(7/2),x]

[Out]

(-2*a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^(5/2)*(a + b*x)) - (2*a^2*(3*A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x)) - (6*a*b*(A*b + a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(Sqrt[x]*(a + b*x)) + (2*b^2*(A*b + 3*a*B)*Sqrt[x
]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*b^3*B*x^(3/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(3*(a + b*x))

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Rubi in Sympy [A]  time = 24.7239, size = 219, normalized size = 1.01 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{5 a x^{\frac{5}{2}}} + \frac{32 b^{2} \sqrt{x} \left (3 A b + 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 \left (a + b x\right )} + \frac{16 b^{2} \sqrt{x} \left (3 A b + 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 a} - \frac{4 b \left (a + b x\right ) \left (3 A b + 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 a \sqrt{x}} - \frac{2 \left (3 A b + 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{15 a x^{\frac{3}{2}}} \]

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)**(3/2)/x**(7/2),x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(5*a*x**(5/2)) + 32*b**2*sq
rt(x)*(3*A*b + 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*(a + b*x)) + 16*b**2*
sqrt(x)*(3*A*b + 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*a) - 4*b*(a + b*x)*
(3*A*b + 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*a*sqrt(x)) - 2*(3*A*b + 5*B*
a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(15*a*x**(3/2))

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Mathematica [A]  time = 0.0684117, size = 83, normalized size = 0.38 \[ -\frac{2 \sqrt{(a+b x)^2} \left (a^3 (3 A+5 B x)+15 a^2 b x (A+3 B x)+45 a b^2 x^2 (A-B x)-5 b^3 x^3 (3 A+B x)\right )}{15 x^{5/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^(7/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(45*a*b^2*x^2*(A - B*x) - 5*b^3*x^3*(3*A + B*x) + 15*a^2*b
*x*(A + 3*B*x) + a^3*(3*A + 5*B*x)))/(15*x^(5/2)*(a + b*x))

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Maple [A]  time = 0.011, size = 92, normalized size = 0.4 \[ -{\frac{-10\,B{x}^{4}{b}^{3}-30\,A{b}^{3}{x}^{3}-90\,B{x}^{3}a{b}^{2}+90\,A{x}^{2}a{b}^{2}+90\,B{x}^{2}{a}^{2}b+30\,A{a}^{2}bx+10\,{a}^{3}Bx+6\,A{a}^{3}}{15\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}} \]

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)^(3/2)/x^(7/2),x)

[Out]

-2/15*(-5*B*b^3*x^4-15*A*b^3*x^3-45*B*a*b^2*x^3+45*A*a*b^2*x^2+45*B*a^2*b*x^2+15
*A*a^2*b*x+5*B*a^3*x+3*A*a^3)*((b*x+a)^2)^(3/2)/x^(5/2)/(b*x+a)^3

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Maxima [A]  time = 0.705469, size = 177, normalized size = 0.82 \[ \frac{2}{3} \, B{\left (\frac{b^{3} x^{2} + 3 \, a b^{2} x}{\sqrt{x}} + \frac{6 \,{\left (a b^{2} x^{2} - a^{2} b x\right )}}{x^{\frac{3}{2}}} - \frac{3 \, a^{2} b x^{2} + a^{3} x}{x^{\frac{5}{2}}}\right )} + \frac{2}{15} \, A{\left (\frac{15 \,{\left (b^{3} x^{2} - a b^{2} x\right )}}{x^{\frac{3}{2}}} - \frac{10 \,{\left (3 \, a b^{2} x^{2} + a^{2} b x\right )}}{x^{\frac{5}{2}}} - \frac{5 \, a^{2} b x^{2} + 3 \, a^{3} x}{x^{\frac{7}{2}}}\right )} \]

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)/x^(7/2),x, algorithm="maxima")

[Out]

2/3*B*((b^3*x^2 + 3*a*b^2*x)/sqrt(x) + 6*(a*b^2*x^2 - a^2*b*x)/x^(3/2) - (3*a^2*
b*x^2 + a^3*x)/x^(5/2)) + 2/15*A*(15*(b^3*x^2 - a*b^2*x)/x^(3/2) - 10*(3*a*b^2*x
^2 + a^2*b*x)/x^(5/2) - (5*a^2*b*x^2 + 3*a^3*x)/x^(7/2))

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Fricas [A]  time = 0.278865, size = 99, normalized size = 0.46 \[ \frac{2 \,{\left (5 \, B b^{3} x^{4} - 3 \, A a^{3} + 15 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} - 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \]

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)/x^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*B*b^3*x^4 - 3*A*a^3 + 15*(3*B*a*b^2 + A*b^3)*x^3 - 45*(B*a^2*b + A*a*b^2
)*x^2 - 5*(B*a^3 + 3*A*a^2*b)*x)/x^(5/2)

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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)**(3/2)/x**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278688, size = 167, normalized size = 0.77 \[ \frac{2}{3} \, B b^{3} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 6 \, B a b^{2} \sqrt{x}{\rm sign}\left (b x + a\right ) + 2 \, A b^{3} \sqrt{x}{\rm sign}\left (b x + a\right ) - \frac{2 \,{\left (45 \, B a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + 45 \, A a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 5 \, B a^{3} x{\rm sign}\left (b x + a\right ) + 15 \, A a^{2} b x{\rm sign}\left (b x + a\right ) + 3 \, A a^{3}{\rm sign}\left (b x + a\right )\right )}}{15 \, x^{\frac{5}{2}}} \]

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)/x^(7/2),x, algorithm="giac")

[Out]

2/3*B*b^3*x^(3/2)*sign(b*x + a) + 6*B*a*b^2*sqrt(x)*sign(b*x + a) + 2*A*b^3*sqrt
(x)*sign(b*x + a) - 2/15*(45*B*a^2*b*x^2*sign(b*x + a) + 45*A*a*b^2*x^2*sign(b*x
+ a) + 5*B*a^3*x*sign(b*x + a) + 15*A*a^2*b*x*sign(b*x + a) + 3*A*a^3*sign(b*x
+ a))/x^(5/2)