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

Optimal. Leaf size=171 \[ \frac{35 b^2 (3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{35 b \sqrt{a+b x} (3 A b-2 a B)}{8 a^5 x}+\frac{35 \sqrt{a+b x} (3 A b-2 a B)}{12 a^4 x^2}-\frac{7 (3 A b-2 a B)}{3 a^3 x^2 \sqrt{a+b x}}-\frac{3 A b-2 a B}{3 a^2 x^2 (a+b x)^{3/2}}-\frac{A}{3 a x^3 (a+b x)^{3/2}} \]

[Out]

-A/(3*a*x^3*(a + b*x)^(3/2)) - (3*A*b - 2*a*B)/(3*a^2*x^2*(a + b*x)^(3/2)) - (7*
(3*A*b - 2*a*B))/(3*a^3*x^2*Sqrt[a + b*x]) + (35*(3*A*b - 2*a*B)*Sqrt[a + b*x])/
(12*a^4*x^2) - (35*b*(3*A*b - 2*a*B)*Sqrt[a + b*x])/(8*a^5*x) + (35*b^2*(3*A*b -
2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(11/2))

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Rubi [A]  time = 0.232797, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{35 b^2 (3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{35 b \sqrt{a+b x} (3 A b-2 a B)}{8 a^5 x}+\frac{35 \sqrt{a+b x} (3 A b-2 a B)}{12 a^4 x^2}-\frac{7 (3 A b-2 a B)}{3 a^3 x^2 \sqrt{a+b x}}-\frac{3 A b-2 a B}{3 a^2 x^2 (a+b x)^{3/2}}-\frac{A}{3 a x^3 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(3*a*x^3*(a + b*x)^(3/2)) - (3*A*b - 2*a*B)/(3*a^2*x^2*(a + b*x)^(3/2)) - (7*
(3*A*b - 2*a*B))/(3*a^3*x^2*Sqrt[a + b*x]) + (35*(3*A*b - 2*a*B)*Sqrt[a + b*x])/
(12*a^4*x^2) - (35*b*(3*A*b - 2*a*B)*Sqrt[a + b*x])/(8*a^5*x) + (35*b^2*(3*A*b -
2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(11/2))

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Rubi in Sympy [A]  time = 22.3841, size = 165, normalized size = 0.96 \[ - \frac{A}{3 a x^{3} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 \left (\frac{3 A b}{2} - B a\right )}{3 a^{2} x^{2} \left (a + b x\right )^{\frac{3}{2}}} - \frac{14 \left (\frac{3 A b}{2} - B a\right )}{3 a^{3} x^{2} \sqrt{a + b x}} + \frac{35 \sqrt{a + b x} \left (\frac{3 A b}{2} - B a\right )}{6 a^{4} x^{2}} - \frac{35 b \sqrt{a + b x} \left (3 A b - 2 B a\right )}{8 a^{5} x} + \frac{35 b^{2} \left (\frac{3 A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(3*a*x**3*(a + b*x)**(3/2)) - 2*(3*A*b/2 - B*a)/(3*a**2*x**2*(a + b*x)**(3/2)
) - 14*(3*A*b/2 - B*a)/(3*a**3*x**2*sqrt(a + b*x)) + 35*sqrt(a + b*x)*(3*A*b/2 -
B*a)/(6*a**4*x**2) - 35*b*sqrt(a + b*x)*(3*A*b - 2*B*a)/(8*a**5*x) + 35*b**2*(3
*A*b/2 - B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(4*a**(11/2))

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Mathematica [A]  time = 0.207227, size = 130, normalized size = 0.76 \[ \frac{35 b^2 (3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{-4 a^4 (2 A+3 B x)+6 a^3 b x (3 A+7 B x)+7 a^2 b^2 x^2 (40 B x-9 A)+210 a b^3 x^3 (B x-2 A)-315 A b^4 x^4}{24 a^5 x^3 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-315*A*b^4*x^4 + 210*a*b^3*x^3*(-2*A + B*x) - 4*a^4*(2*A + 3*B*x) + 6*a^3*b*x*(
3*A + 7*B*x) + 7*a^2*b^2*x^2*(-9*A + 40*B*x))/(24*a^5*x^3*(a + b*x)^(3/2)) + (35
*b^2*(3*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(11/2))

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Maple [A]  time = 0.026, size = 147, normalized size = 0.9 \[ 2\,{b}^{2} \left ( -1/3\,{\frac{Ab-Ba}{{a}^{4} \left ( bx+a \right ) ^{3/2}}}-{\frac{4\,Ab-3\,Ba}{{a}^{5}\sqrt{bx+a}}}-{\frac{1}{{a}^{5}} \left ({\frac{1}{{x}^{3}{b}^{3}} \left ( \left ({\frac{41\,Ab}{16}}-{\frac{11\,Ba}{8}} \right ) \left ( bx+a \right ) ^{5/2}+ \left ( -{\frac{35\,Aab}{6}}+3\,B{a}^{2} \right ) \left ( bx+a \right ) ^{3/2}+ \left ({\frac{55\,A{a}^{2}b}{16}}-{\frac{13\,B{a}^{3}}{8}} \right ) \sqrt{bx+a} \right ) }-{\frac{105\,Ab-70\,Ba}{16\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^2*(-1/3*(A*b-B*a)/a^4/(b*x+a)^(3/2)-(4*A*b-3*B*a)/a^5/(b*x+a)^(1/2)-1/a^5*((
(41/16*A*b-11/8*B*a)*(b*x+a)^(5/2)+(-35/6*A*a*b+3*B*a^2)*(b*x+a)^(3/2)+(55/16*A*
a^2*b-13/8*B*a^3)*(b*x+a)^(1/2))/x^3/b^3-35/16*(3*A*b-2*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 = 0.239357, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left ({\left (2 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} +{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3}\right )} \sqrt{b x + a} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (8 \, A a^{4} - 105 \,{\left (2 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} - 140 \,{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3} - 21 \,{\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} + 6 \,{\left (2 \, B a^{4} - 3 \, A a^{3} b\right )} x\right )} \sqrt{a}}{48 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )} \sqrt{b x + a} \sqrt{a}}, \frac{105 \,{\left ({\left (2 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} +{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3}\right )} \sqrt{b x + a} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (8 \, A a^{4} - 105 \,{\left (2 \, B a b^{3} - 3 \, A b^{4}\right )} x^{4} - 140 \,{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3} - 21 \,{\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} + 6 \,{\left (2 \, B a^{4} - 3 \, A a^{3} b\right )} x\right )} \sqrt{-a}}{24 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )} \sqrt{b x + a} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(105*((2*B*a*b^3 - 3*A*b^4)*x^4 + (2*B*a^2*b^2 - 3*A*a*b^3)*x^3)*sqrt(b*x
+ a)*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) + 2*(8*A*a^4 - 105*(2*B*a
*b^3 - 3*A*b^4)*x^4 - 140*(2*B*a^2*b^2 - 3*A*a*b^3)*x^3 - 21*(2*B*a^3*b - 3*A*a^
2*b^2)*x^2 + 6*(2*B*a^4 - 3*A*a^3*b)*x)*sqrt(a))/((a^5*b*x^4 + a^6*x^3)*sqrt(b*x
+ a)*sqrt(a)), 1/24*(105*((2*B*a*b^3 - 3*A*b^4)*x^4 + (2*B*a^2*b^2 - 3*A*a*b^3)
*x^3)*sqrt(b*x + a)*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - (8*A*a^4 - 105*(2*B*a*b
^3 - 3*A*b^4)*x^4 - 140*(2*B*a^2*b^2 - 3*A*a*b^3)*x^3 - 21*(2*B*a^3*b - 3*A*a^2*
b^2)*x^2 + 6*(2*B*a^4 - 3*A*a^3*b)*x)*sqrt(-a))/((a^5*b*x^4 + a^6*x^3)*sqrt(b*x
+ a)*sqrt(-a))]

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Sympy [A]  time = 86.1522, size = 1001, normalized size = 5.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-8*a**(133/2)*b**128*x**128/(24*a**(137/2)*b**(257/2)*x**(263/2)*sqrt(a/(b*x)
+ 1) + 24*a**(135/2)*b**(259/2)*x**(265/2)*sqrt(a/(b*x) + 1)) + 18*a**(131/2)*b
**129*x**129/(24*a**(137/2)*b**(257/2)*x**(263/2)*sqrt(a/(b*x) + 1) + 24*a**(135
/2)*b**(259/2)*x**(265/2)*sqrt(a/(b*x) + 1)) - 63*a**(129/2)*b**130*x**130/(24*a
**(137/2)*b**(257/2)*x**(263/2)*sqrt(a/(b*x) + 1) + 24*a**(135/2)*b**(259/2)*x**
(265/2)*sqrt(a/(b*x) + 1)) - 420*a**(127/2)*b**131*x**131/(24*a**(137/2)*b**(257
/2)*x**(263/2)*sqrt(a/(b*x) + 1) + 24*a**(135/2)*b**(259/2)*x**(265/2)*sqrt(a/(b
*x) + 1)) - 315*a**(125/2)*b**132*x**132/(24*a**(137/2)*b**(257/2)*x**(263/2)*sq
rt(a/(b*x) + 1) + 24*a**(135/2)*b**(259/2)*x**(265/2)*sqrt(a/(b*x) + 1)) + 315*a
**63*b**(263/2)*x**(263/2)*sqrt(a/(b*x) + 1)*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(2
4*a**(137/2)*b**(257/2)*x**(263/2)*sqrt(a/(b*x) + 1) + 24*a**(135/2)*b**(259/2)*
x**(265/2)*sqrt(a/(b*x) + 1)) + 315*a**62*b**(265/2)*x**(265/2)*sqrt(a/(b*x) + 1
)*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(24*a**(137/2)*b**(257/2)*x**(263/2)*sqrt(a/(
b*x) + 1) + 24*a**(135/2)*b**(259/2)*x**(265/2)*sqrt(a/(b*x) + 1))) + B*(-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**(153/2)*x**(1
57/2)*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)*sqrt(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)*sqr
t(a/(b*x) + 1)) - 105*a**41*b**(157/2)*x**(157/2)*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)))

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GIAC/XCAS [A]  time = 0.218604, size = 270, normalized size = 1.58 \[ \frac{35 \,{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{5}} + \frac{210 \,{\left (b x + a\right )}^{4} B a b^{2} - 560 \,{\left (b x + a\right )}^{3} B a^{2} b^{2} + 462 \,{\left (b x + a\right )}^{2} B a^{3} b^{2} - 96 \,{\left (b x + a\right )} B a^{4} b^{2} - 16 \, B a^{5} b^{2} - 315 \,{\left (b x + a\right )}^{4} A b^{3} + 840 \,{\left (b x + a\right )}^{3} A a b^{3} - 693 \,{\left (b x + a\right )}^{2} A a^{2} b^{3} + 144 \,{\left (b x + a\right )} A a^{3} b^{3} + 16 \, A a^{4} b^{3}}{24 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - \sqrt{b x + a} a\right )}^{3} a^{5}} \]

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]

35/8*(2*B*a*b^2 - 3*A*b^3)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^5) + 1/24*
(210*(b*x + a)^4*B*a*b^2 - 560*(b*x + a)^3*B*a^2*b^2 + 462*(b*x + a)^2*B*a^3*b^2
- 96*(b*x + a)*B*a^4*b^2 - 16*B*a^5*b^2 - 315*(b*x + a)^4*A*b^3 + 840*(b*x + a)
^3*A*a*b^3 - 693*(b*x + a)^2*A*a^2*b^3 + 144*(b*x + a)*A*a^3*b^3 + 16*A*a^4*b^3)
/(((b*x + a)^(3/2) - sqrt(b*x + a)*a)^3*a^5)