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

Optimal. Leaf size=140 \[ -\frac{5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{5 \sqrt{a+b x} (7 A b-4 a B)}{4 a^4 x}-\frac{5 (7 A b-4 a B)}{6 a^3 x \sqrt{a+b x}}-\frac{7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac{A}{2 a x^2 (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.183355, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{5 \sqrt{a+b x} (7 A b-4 a B)}{4 a^4 x}-\frac{5 (7 A b-4 a B)}{6 a^3 x \sqrt{a+b x}}-\frac{7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac{A}{2 a x^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.9104, size = 129, normalized size = 0.92 \[ - \frac{A}{2 a x^{2} \left (a + b x\right )^{\frac{3}{2}}} - \frac{7 A b - 4 B a}{6 a^{2} x \left (a + b x\right )^{\frac{3}{2}}} - \frac{5 \left (7 A b - 4 B a\right )}{6 a^{3} x \sqrt{a + b x}} + \frac{5 \sqrt{a + b x} \left (7 A b - 4 B a\right )}{4 a^{4} x} - \frac{5 b \left (7 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(2*a*x**2*(a + b*x)**(3/2)) - (7*A*b - 4*B*a)/(6*a**2*x*(a + b*x)**(3/2)) - 5
*(7*A*b - 4*B*a)/(6*a**3*x*sqrt(a + b*x)) + 5*sqrt(a + b*x)*(7*A*b - 4*B*a)/(4*a
**4*x) - 5*b*(7*A*b - 4*B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(4*a**(9/2))

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Mathematica [A]  time = 0.171861, size = 107, normalized size = 0.76 \[ \frac{5 b (4 a B-7 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{-6 a^3 (A+2 B x)+a^2 b x (21 A-80 B x)+20 a b^2 x^2 (7 A-3 B x)+105 A b^3 x^3}{12 a^4 x^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(105*A*b^3*x^3 + a^2*b*x*(21*A - 80*B*x) + 20*a*b^2*x^2*(7*A - 3*B*x) - 6*a^3*(A
+ 2*B*x))/(12*a^4*x^2*(a + b*x)^(3/2)) + (5*b*(-7*A*b + 4*a*B)*ArcTanh[Sqrt[a +
b*x]/Sqrt[a]])/(4*a^(9/2))

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Maple [A]  time = 0.024, size = 122, normalized size = 0.9 \[ 2\,b \left ( -{\frac{-3\,Ab+2\,Ba}{{a}^{4}\sqrt{bx+a}}}-1/3\,{\frac{-Ab+Ba}{{a}^{3} \left ( bx+a \right ) ^{3/2}}}+{\frac{1}{{a}^{4}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( \left ({\frac{11\,Ab}{8}}-1/2\,Ba \right ) \left ( bx+a \right ) ^{3/2}+ \left ( -{\frac{13\,Aab}{8}}+1/2\,B{a}^{2} \right ) \sqrt{bx+a} \right ) }-5/8\,{\frac{7\,Ab-4\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*(-(-3*A*b+2*B*a)/a^4/(b*x+a)^(1/2)-1/3*(-A*b+B*a)/a^3/(b*x+a)^(3/2)+1/a^4*((
(11/8*A*b-1/2*B*a)*(b*x+a)^(3/2)+(-13/8*A*a*b+1/2*B*a^2)*(b*x+a)^(1/2))/x^2/b^2-
5/8*(7*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 = 0.235656, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left ({\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} +{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x + a} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (6 \, A a^{3} + 15 \,{\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 20 \,{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 3 \,{\left (4 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{a}}{24 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )} \sqrt{b x + a} \sqrt{a}}, -\frac{15 \,{\left ({\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} +{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x + a} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (6 \, A a^{3} + 15 \,{\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 20 \,{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 3 \,{\left (4 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{-a}}{12 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )} \sqrt{b x + a} \sqrt{-a}}\right ] \]

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^2 - 7*A*b^3)*x^3 + (4*B*a^2*b - 7*A*a*b^2)*x^2)*sqrt(b*x +
a)*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) + 2*(6*A*a^3 + 15*(4*B*a*b^2
- 7*A*b^3)*x^3 + 20*(4*B*a^2*b - 7*A*a*b^2)*x^2 + 3*(4*B*a^3 - 7*A*a^2*b)*x)*sq
rt(a))/((a^4*b*x^3 + a^5*x^2)*sqrt(b*x + a)*sqrt(a)), -1/12*(15*((4*B*a*b^2 - 7*
A*b^3)*x^3 + (4*B*a^2*b - 7*A*a*b^2)*x^2)*sqrt(b*x + a)*arctan(a/(sqrt(b*x + a)*
sqrt(-a))) + (6*A*a^3 + 15*(4*B*a*b^2 - 7*A*b^3)*x^3 + 20*(4*B*a^2*b - 7*A*a*b^2
)*x^2 + 3*(4*B*a^3 - 7*A*a^2*b)*x)*sqrt(-a))/((a^4*b*x^3 + a^5*x^2)*sqrt(b*x + a
)*sqrt(-a))]

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Sympy [A]  time = 58.8049, size = 1287, normalized size = 9.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-6*a**(89/2)*b**75*x**75/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) + 1
) + 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) + 1)) + 21*a**(87/2)*b**76*x
**76/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) + 1) + 12*a**(91/2)*b**(15
3/2)*x**(157/2)*sqrt(a/(b*x) + 1)) + 140*a**(85/2)*b**77*x**77/(12*a**(93/2)*b**
(151/2)*x**(155/2)*sqrt(a/(b*x) + 1) + 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a
/(b*x) + 1)) + 105*a**(83/2)*b**78*x**78/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqr
t(a/(b*x) + 1) + 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) + 1)) - 105*a**
42*b**(155/2)*x**(155/2)*sqrt(a/(b*x) + 1)*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(12*
a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) + 1) + 12*a**(91/2)*b**(153/2)*x**(
157/2)*sqrt(a/(b*x) + 1)) - 105*a**41*b**(157/2)*x**(157/2)*sqrt(a/(b*x) + 1)*as
inh(sqrt(a)/(sqrt(b)*sqrt(x)))/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x)
+ 1) + 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) + 1))) + B*(-6*a**17*sqrt
(1 + b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a*
*(33/2)*b**3*x**4) - 46*a**16*b*x*sqrt(1 + b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*
b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 15*a**16*b*x*log(b*x/
a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b
**3*x**4) + 30*a**16*b*x*log(sqrt(1 + b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*
b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 70*a**15*b**2*x**2*sq
rt(1 + b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*
a**(33/2)*b**3*x**4) - 45*a**15*b**2*x**2*log(b*x/a)/(6*a**(39/2)*x + 18*a**(37/
2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 90*a**15*b**2*x**2
*log(sqrt(1 + b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b*
*2*x**3 + 6*a**(33/2)*b**3*x**4) - 30*a**14*b**3*x**3*sqrt(1 + b*x/a)/(6*a**(39/
2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 4
5*a**14*b**3*x**3*log(b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)
*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 90*a**14*b**3*x**3*log(sqrt(1 + b*x/a) + 1
)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b*
*3*x**4) - 15*a**13*b**4*x**4*log(b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 +
18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*a**13*b**4*x**4*log(sqrt(1
+ b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*
a**(33/2)*b**3*x**4))

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GIAC/XCAS [A]  time = 0.217608, size = 201, normalized size = 1.44 \[ -\frac{5 \,{\left (4 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{4}} - \frac{2 \,{\left (6 \,{\left (b x + a\right )} B a b + B a^{2} b - 9 \,{\left (b x + a\right )} A b^{2} - A a b^{2}\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}} - \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b - 4 \, \sqrt{b x + a} B a^{2} b - 11 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{2} + 13 \, \sqrt{b x + a} A a b^{2}}{4 \, a^{4} b^{2} x^{2}} \]

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]

-5/4*(4*B*a*b - 7*A*b^2)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) - 2/3*(6*
(b*x + a)*B*a*b + B*a^2*b - 9*(b*x + a)*A*b^2 - A*a*b^2)/((b*x + a)^(3/2)*a^4) -
1/4*(4*(b*x + a)^(3/2)*B*a*b - 4*sqrt(b*x + a)*B*a^2*b - 11*(b*x + a)^(3/2)*A*b
^2 + 13*sqrt(b*x + a)*A*a*b^2)/(a^4*b^2*x^2)