3.406 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^4} \, dx\)

Optimal. Leaf size=139 \[ \frac{5 b^2 \sqrt{a+b x} (6 a B+A b)}{8 a}-\frac{5 b^2 (6 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{(a+b x)^{5/2} (6 a B+A b)}{12 a x^2}-\frac{5 b (a+b x)^{3/2} (6 a B+A b)}{24 a x}-\frac{A (a+b x)^{7/2}}{3 a x^3} \]

[Out]

(5*b^2*(A*b + 6*a*B)*Sqrt[a + b*x])/(8*a) - (5*b*(A*b + 6*a*B)*(a + b*x)^(3/2))/
(24*a*x) - ((A*b + 6*a*B)*(a + b*x)^(5/2))/(12*a*x^2) - (A*(a + b*x)^(7/2))/(3*a
*x^3) - (5*b^2*(A*b + 6*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*Sqrt[a])

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Rubi [A]  time = 0.173162, antiderivative size = 139, 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 \[ \frac{5 b^2 \sqrt{a+b x} (6 a B+A b)}{8 a}-\frac{5 b^2 (6 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{(a+b x)^{5/2} (6 a B+A b)}{12 a x^2}-\frac{5 b (a+b x)^{3/2} (6 a B+A b)}{24 a x}-\frac{A (a+b x)^{7/2}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

(5*b^2*(A*b + 6*a*B)*Sqrt[a + b*x])/(8*a) - (5*b*(A*b + 6*a*B)*(a + b*x)^(3/2))/
(24*a*x) - ((A*b + 6*a*B)*(a + b*x)^(5/2))/(12*a*x^2) - (A*(a + b*x)^(7/2))/(3*a
*x^3) - (5*b^2*(A*b + 6*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*Sqrt[a])

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Rubi in Sympy [A]  time = 15.5618, size = 126, normalized size = 0.91 \[ - \frac{A \left (a + b x\right )^{\frac{7}{2}}}{3 a x^{3}} + \frac{5 b^{2} \sqrt{a + b x} \left (A b + 6 B a\right )}{8 a} - \frac{5 b \left (a + b x\right )^{\frac{3}{2}} \left (A b + 6 B a\right )}{24 a x} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (A b + 6 B a\right )}{12 a x^{2}} - \frac{5 b^{2} \left (A b + 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(a + b*x)**(7/2)/(3*a*x**3) + 5*b**2*sqrt(a + b*x)*(A*b + 6*B*a)/(8*a) - 5*b*
(a + b*x)**(3/2)*(A*b + 6*B*a)/(24*a*x) - (a + b*x)**(5/2)*(A*b + 6*B*a)/(12*a*x
**2) - 5*b**2*(A*b + 6*B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(8*sqrt(a))

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Mathematica [A]  time = 0.179172, size = 96, normalized size = 0.69 \[ -\frac{\sqrt{a+b x} \left (4 a^2 (2 A+3 B x)+2 a b x (13 A+27 B x)+3 b^2 x^2 (11 A-16 B x)\right )}{24 x^3}-\frac{5 b^2 (6 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]*(3*b^2*x^2*(11*A - 16*B*x) + 4*a^2*(2*A + 3*B*x) + 2*a*b*x*(13*A
+ 27*B*x)))/(24*x^3) - (5*b^2*(A*b + 6*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*
Sqrt[a])

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Maple [A]  time = 0.02, size = 108, normalized size = 0.8 \[ 2\,{b}^{2} \left ( B\sqrt{bx+a}+{\frac{1}{{x}^{3}{b}^{3}} \left ( \left ( -{\frac{11\,Ab}{16}}-{\frac{9\,Ba}{8}} \right ) \left ( bx+a \right ) ^{5/2}+ \left ( 5/6\,Aab+2\,B{a}^{2} \right ) \left ( bx+a \right ) ^{3/2}+ \left ( -{\frac{7\,B{a}^{3}}{8}}-{\frac{5\,A{a}^{2}b}{16}} \right ) \sqrt{bx+a} \right ) }-{\frac{5\,Ab+30\,Ba}{16\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^2*(B*(b*x+a)^(1/2)+((-11/16*A*b-9/8*B*a)*(b*x+a)^(5/2)+(5/6*A*a*b+2*B*a^2)*(
b*x+a)^(3/2)+(-7/8*B*a^3-5/16*A*a^2*b)*(b*x+a)^(1/2))/x^3/b^3-5/16*(A*b+6*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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222134, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (6 \, B a b^{2} + A b^{3}\right )} x^{3} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (48 \, B b^{2} x^{3} - 8 \, A a^{2} - 3 \,{\left (18 \, B a b + 11 \, A b^{2}\right )} x^{2} - 2 \,{\left (6 \, B a^{2} + 13 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{48 \, \sqrt{a} x^{3}}, \frac{15 \,{\left (6 \, B a b^{2} + A b^{3}\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (48 \, B b^{2} x^{3} - 8 \, A a^{2} - 3 \,{\left (18 \, B a b + 11 \, A b^{2}\right )} x^{2} - 2 \,{\left (6 \, B a^{2} + 13 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{24 \, \sqrt{-a} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*(6*B*a*b^2 + A*b^3)*x^3*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/
x) + 2*(48*B*b^2*x^3 - 8*A*a^2 - 3*(18*B*a*b + 11*A*b^2)*x^2 - 2*(6*B*a^2 + 13*A
*a*b)*x)*sqrt(b*x + a)*sqrt(a))/(sqrt(a)*x^3), 1/24*(15*(6*B*a*b^2 + A*b^3)*x^3*
arctan(a/(sqrt(b*x + a)*sqrt(-a))) + (48*B*b^2*x^3 - 8*A*a^2 - 3*(18*B*a*b + 11*
A*b^2)*x^2 - 2*(6*B*a^2 + 13*A*a*b)*x)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*x^3)]

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Sympy [A]  time = 90.7369, size = 989, normalized size = 7.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-66*A*a**5*b**3*sqrt(a + b*x)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 +
48*a**3*(a + b*x)**3) + 80*A*a**4*b**3*(a + b*x)**(3/2)/(96*a**6 + 144*a**5*b*x
- 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) - 30*A*a**3*b**3*(a + b*x)**(5/2
)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) - 30*A
*a**3*b**3*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) - 5*A*a**
3*b**3*sqrt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 5*A*a**3*b**3
*sqrt(a**(-7))*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 18*A*a**2*b**3*(a +
b*x)**(3/2)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 9*A*a**2*b**3*sqrt(a
**(-5))*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 9*A*a**2*b**3*sqrt(a**(-5))
*log(a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 3*A*a*b**3*sqrt(a**(-3))*log(-a**2*
sqrt(a**(-3)) + sqrt(a + b*x))/2 + 3*A*a*b**3*sqrt(a**(-3))*log(a**2*sqrt(a**(-3
)) + sqrt(a + b*x))/2 - 2*A*b**3*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*b**2*sqrt
(a + b*x)/x - 10*B*a**4*b**2*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a +
b*x)**2) + 6*B*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*B*a**3*b**2*sqrt(a**(-5))*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8
- 3*B*a**3*b**2*sqrt(a**(-5))*log(a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 3*B*a
**2*b**2*sqrt(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 + 3*B*a**2*b**
2*sqrt(a**(-3))*log(a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 - 6*B*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*B*a*b*sqrt(a + b*x)/x + 2*B*b**2*sqrt(a + b*x)

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GIAC/XCAS [A]  time = 0.226386, size = 204, normalized size = 1.47 \[ \frac{48 \, \sqrt{b x + a} B b^{3} + \frac{15 \,{\left (6 \, B a b^{3} + A b^{4}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{54 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{3} - 96 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{3} + 42 \, \sqrt{b x + a} B a^{3} b^{3} + 33 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{4} - 40 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{4} + 15 \, \sqrt{b x + a} A a^{2} b^{4}}{b^{3} x^{3}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(48*sqrt(b*x + a)*B*b^3 + 15*(6*B*a*b^3 + A*b^4)*arctan(sqrt(b*x + a)/sqrt(
-a))/sqrt(-a) - (54*(b*x + a)^(5/2)*B*a*b^3 - 96*(b*x + a)^(3/2)*B*a^2*b^3 + 42*
sqrt(b*x + a)*B*a^3*b^3 + 33*(b*x + a)^(5/2)*A*b^4 - 40*(b*x + a)^(3/2)*A*a*b^4
+ 15*sqrt(b*x + a)*A*a^2*b^4)/(b^3*x^3))/b