3.405 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^3} \, dx\)

Optimal. Leaf size=133 \[ -\frac{(a+b x)^{5/2} (4 a B+3 A b)}{4 a x}+\frac{5 b (a+b x)^{3/2} (4 a B+3 A b)}{12 a}+\frac{5}{4} b \sqrt{a+b x} (4 a B+3 A b)-\frac{5}{4} \sqrt{a} b (4 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{A (a+b x)^{7/2}}{2 a x^2} \]

[Out]

(5*b*(3*A*b + 4*a*B)*Sqrt[a + b*x])/4 + (5*b*(3*A*b + 4*a*B)*(a + b*x)^(3/2))/(1
2*a) - ((3*A*b + 4*a*B)*(a + b*x)^(5/2))/(4*a*x) - (A*(a + b*x)^(7/2))/(2*a*x^2)
- (5*Sqrt[a]*b*(3*A*b + 4*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/4

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Rubi [A]  time = 0.166418, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{(a+b x)^{5/2} (4 a B+3 A b)}{4 a x}+\frac{5 b (a+b x)^{3/2} (4 a B+3 A b)}{12 a}+\frac{5}{4} b \sqrt{a+b x} (4 a B+3 A b)-\frac{5}{4} \sqrt{a} b (4 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{A (a+b x)^{7/2}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^3,x]

[Out]

(5*b*(3*A*b + 4*a*B)*Sqrt[a + b*x])/4 + (5*b*(3*A*b + 4*a*B)*(a + b*x)^(3/2))/(1
2*a) - ((3*A*b + 4*a*B)*(a + b*x)^(5/2))/(4*a*x) - (A*(a + b*x)^(7/2))/(2*a*x^2)
- (5*Sqrt[a]*b*(3*A*b + 4*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/4

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Rubi in Sympy [A]  time = 14.9733, size = 124, normalized size = 0.93 \[ - \frac{A \left (a + b x\right )^{\frac{7}{2}}}{2 a x^{2}} - \frac{5 \sqrt{a} b \left (3 A b + 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4} + \frac{5 b \sqrt{a + b x} \left (3 A b + 4 B a\right )}{4} + \frac{5 b \left (a + b x\right )^{\frac{3}{2}} \left (3 A b + 4 B a\right )}{12 a} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (3 A b + 4 B a\right )}{4 a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**3,x)

[Out]

-A*(a + b*x)**(7/2)/(2*a*x**2) - 5*sqrt(a)*b*(3*A*b + 4*B*a)*atanh(sqrt(a + b*x)
/sqrt(a))/4 + 5*b*sqrt(a + b*x)*(3*A*b + 4*B*a)/4 + 5*b*(a + b*x)**(3/2)*(3*A*b
+ 4*B*a)/(12*a) - (a + b*x)**(5/2)*(3*A*b + 4*B*a)/(4*a*x)

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Mathematica [A]  time = 0.1485, size = 91, normalized size = 0.68 \[ \frac{\sqrt{a+b x} \left (-6 a^2 (A+2 B x)+a b x (56 B x-27 A)+8 b^2 x^2 (3 A+B x)\right )}{12 x^2}-\frac{5}{4} \sqrt{a} b (4 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^3,x]

[Out]

(Sqrt[a + b*x]*(8*b^2*x^2*(3*A + B*x) - 6*a^2*(A + 2*B*x) + a*b*x*(-27*A + 56*B*
x)))/(12*x^2) - (5*Sqrt[a]*b*(3*A*b + 4*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/4

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Maple [A]  time = 0.019, size = 110, normalized size = 0.8 \[ 2\,b \left ( 1/3\,B \left ( bx+a \right ) ^{3/2}+Ab\sqrt{bx+a}+2\,Ba\sqrt{bx+a}+a \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( \left ( -{\frac{9\,Ab}{8}}-1/2\,Ba \right ) \left ( bx+a \right ) ^{3/2}+ \left ({\frac{7\,Aab}{8}}+1/2\,B{a}^{2} \right ) \sqrt{bx+a} \right ) }-5/8\,{\frac{3\,Ab+4\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^3,x)

[Out]

2*b*(1/3*B*(b*x+a)^(3/2)+A*b*(b*x+a)^(1/2)+2*B*a*(b*x+a)^(1/2)+a*(((-9/8*A*b-1/2
*B*a)*(b*x+a)^(3/2)+(7/8*A*a*b+1/2*B*a^2)*(b*x+a)^(1/2))/x^2/b^2-5/8*(3*A*b+4*B*
a)/a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2))))

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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*x + A)*(b*x + a)^(5/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223489, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (4 \, B a b + 3 \, A b^{2}\right )} \sqrt{a} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (8 \, B b^{2} x^{3} - 6 \, A a^{2} + 8 \,{\left (7 \, B a b + 3 \, A b^{2}\right )} x^{2} - 3 \,{\left (4 \, B a^{2} + 9 \, A a b\right )} x\right )} \sqrt{b x + a}}{24 \, x^{2}}, -\frac{15 \,{\left (4 \, B a b + 3 \, A b^{2}\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (8 \, B b^{2} x^{3} - 6 \, A a^{2} + 8 \,{\left (7 \, B a b + 3 \, A b^{2}\right )} x^{2} - 3 \,{\left (4 \, B a^{2} + 9 \, A a b\right )} x\right )} \sqrt{b x + a}}{12 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^3,x, algorithm="fricas")

[Out]

[1/24*(15*(4*B*a*b + 3*A*b^2)*sqrt(a)*x^2*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2
*a)/x) + 2*(8*B*b^2*x^3 - 6*A*a^2 + 8*(7*B*a*b + 3*A*b^2)*x^2 - 3*(4*B*a^2 + 9*A
*a*b)*x)*sqrt(b*x + a))/x^2, -1/12*(15*(4*B*a*b + 3*A*b^2)*sqrt(-a)*x^2*arctan(s
qrt(b*x + a)/sqrt(-a)) - (8*B*b^2*x^3 - 6*A*a^2 + 8*(7*B*a*b + 3*A*b^2)*x^2 - 3*
(4*B*a^2 + 9*A*a*b)*x)*sqrt(b*x + a))/x^2]

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Sympy [A]  time = 59.3872, size = 600, normalized size = 4.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)*(B*x+A)/x**3,x)

[Out]

-10*A*a**4*b**2*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 6*
A*a**3*b**2*(a + b*x)**(3/2)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 3*A
*a**3*b**2*sqrt(a**(-5))*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 3*A*a**3*b
**2*sqrt(a**(-5))*log(a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 3*A*a**2*b**2*sqrt
(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 + 3*A*a**2*b**2*sqrt(a**(-3
))*log(a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 - 6*A*a*b**2*Piecewise((-atan(sqrt(
a + b*x)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a
< 0) & (a < a + b*x)), (atanh(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (a > a
+ b*x))) - 3*A*a*b*sqrt(a + b*x)/x + 2*A*b**2*sqrt(a + b*x) - B*a**3*b*sqrt(a**
(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 + B*a**3*b*sqrt(a**(-3))*log(a*
*2*sqrt(a**(-3)) + sqrt(a + b*x))/2 - 6*B*a**2*b*Piecewise((-atan(sqrt(a + b*x)/
sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (
a < a + b*x)), (atanh(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (a > a + b*x)))
- B*a**2*sqrt(a + b*x)/x + 4*B*a*b*sqrt(a + b*x) + B*b**2*Piecewise((sqrt(a)*x,
Eq(b, 0)), (2*(a + b*x)**(3/2)/(3*b), True))

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GIAC/XCAS [A]  time = 0.224622, size = 209, normalized size = 1.57 \[ \frac{8 \,{\left (b x + a\right )}^{\frac{3}{2}} B b^{2} + 48 \, \sqrt{b x + a} B a b^{2} + 24 \, \sqrt{b x + a} A b^{3} + \frac{15 \,{\left (4 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{3 \,{\left (4 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{2} - 4 \, \sqrt{b x + a} B a^{3} b^{2} + 9 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{3} - 7 \, \sqrt{b x + a} A a^{2} b^{3}\right )}}{b^{2} x^{2}}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^3,x, algorithm="giac")

[Out]

1/12*(8*(b*x + a)^(3/2)*B*b^2 + 48*sqrt(b*x + a)*B*a*b^2 + 24*sqrt(b*x + a)*A*b^
3 + 15*(4*B*a^2*b^2 + 3*A*a*b^3)*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) - 3*(4*
(b*x + a)^(3/2)*B*a^2*b^2 - 4*sqrt(b*x + a)*B*a^3*b^2 + 9*(b*x + a)^(3/2)*A*a*b^
3 - 7*sqrt(b*x + a)*A*a^2*b^3)/(b^2*x^2))/b