3.532 \(\int \frac{A+B x}{x^{7/2} (a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{32 b \sqrt{a+b x} (8 A b-5 a B)}{15 a^5 \sqrt{x}}+\frac{16 \sqrt{a+b x} (8 A b-5 a B)}{15 a^4 x^{3/2}}-\frac{4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt{a+b x}}-\frac{2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac{2 A}{5 a x^{5/2} (a+b x)^{3/2}} \]

[Out]

(-2*A)/(5*a*x^(5/2)*(a + b*x)^(3/2)) - (2*(8*A*b - 5*a*B))/(15*a^2*x^(3/2)*(a +
b*x)^(3/2)) - (4*(8*A*b - 5*a*B))/(5*a^3*x^(3/2)*Sqrt[a + b*x]) + (16*(8*A*b - 5
*a*B)*Sqrt[a + b*x])/(15*a^4*x^(3/2)) - (32*b*(8*A*b - 5*a*B)*Sqrt[a + b*x])/(15
*a^5*Sqrt[x])

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Rubi [A]  time = 0.166089, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{32 b \sqrt{a+b x} (8 A b-5 a B)}{15 a^5 \sqrt{x}}+\frac{16 \sqrt{a+b x} (8 A b-5 a B)}{15 a^4 x^{3/2}}-\frac{4 (8 A b-5 a B)}{5 a^3 x^{3/2} \sqrt{a+b x}}-\frac{2 (8 A b-5 a B)}{15 a^2 x^{3/2} (a+b x)^{3/2}}-\frac{2 A}{5 a x^{5/2} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(7/2)*(a + b*x)^(5/2)),x]

[Out]

(-2*A)/(5*a*x^(5/2)*(a + b*x)^(3/2)) - (2*(8*A*b - 5*a*B))/(15*a^2*x^(3/2)*(a +
b*x)^(3/2)) - (4*(8*A*b - 5*a*B))/(5*a^3*x^(3/2)*Sqrt[a + b*x]) + (16*(8*A*b - 5
*a*B)*Sqrt[a + b*x])/(15*a^4*x^(3/2)) - (32*b*(8*A*b - 5*a*B)*Sqrt[a + b*x])/(15
*a^5*Sqrt[x])

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Rubi in Sympy [A]  time = 14.4117, size = 139, normalized size = 0.97 \[ - \frac{2 A}{5 a x^{\frac{5}{2}} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 \left (8 A b - 5 B a\right )}{15 a^{2} x^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}}} - \frac{4 \left (8 A b - 5 B a\right )}{5 a^{3} x^{\frac{3}{2}} \sqrt{a + b x}} + \frac{16 \sqrt{a + b x} \left (8 A b - 5 B a\right )}{15 a^{4} x^{\frac{3}{2}}} - \frac{32 b \sqrt{a + b x} \left (8 A b - 5 B a\right )}{15 a^{5} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(7/2)/(b*x+a)**(5/2),x)

[Out]

-2*A/(5*a*x**(5/2)*(a + b*x)**(3/2)) - 2*(8*A*b - 5*B*a)/(15*a**2*x**(3/2)*(a +
b*x)**(3/2)) - 4*(8*A*b - 5*B*a)/(5*a**3*x**(3/2)*sqrt(a + b*x)) + 16*sqrt(a + b
*x)*(8*A*b - 5*B*a)/(15*a**4*x**(3/2)) - 32*b*sqrt(a + b*x)*(8*A*b - 5*B*a)/(15*
a**5*sqrt(x))

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Mathematica [A]  time = 0.119848, size = 94, normalized size = 0.65 \[ -\frac{2 \left (a^4 (3 A+5 B x)-2 a^3 b x (4 A+15 B x)+24 a^2 b^2 x^2 (2 A-5 B x)+16 a b^3 x^3 (12 A-5 B x)+128 A b^4 x^4\right )}{15 a^5 x^{5/2} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(7/2)*(a + b*x)^(5/2)),x]

[Out]

(-2*(128*A*b^4*x^4 + 24*a^2*b^2*x^2*(2*A - 5*B*x) + 16*a*b^3*x^3*(12*A - 5*B*x)
+ a^4*(3*A + 5*B*x) - 2*a^3*b*x*(4*A + 15*B*x)))/(15*a^5*x^(5/2)*(a + b*x)^(3/2)
)

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Maple [A]  time = 0.008, size = 101, normalized size = 0.7 \[ -{\frac{256\,A{b}^{4}{x}^{4}-160\,Ba{b}^{3}{x}^{4}+384\,Aa{b}^{3}{x}^{3}-240\,B{a}^{2}{b}^{2}{x}^{3}+96\,A{a}^{2}{b}^{2}{x}^{2}-60\,B{a}^{3}b{x}^{2}-16\,A{a}^{3}bx+10\,B{a}^{4}x+6\,A{a}^{4}}{15\,{a}^{5}}{x}^{-{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(7/2)/(b*x+a)^(5/2),x)

[Out]

-2/15*(128*A*b^4*x^4-80*B*a*b^3*x^4+192*A*a*b^3*x^3-120*B*a^2*b^2*x^3+48*A*a^2*b
^2*x^2-30*B*a^3*b*x^2-8*A*a^3*b*x+5*B*a^4*x+3*A*a^4)/x^(5/2)/(b*x+a)^(3/2)/a^5

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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^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244121, size = 157, normalized size = 1.09 \[ -\frac{2 \,{\left (3 \, A a^{4} - 16 \,{\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} - 24 \,{\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 6 \,{\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} +{\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )}}{15 \,{\left (a^{5} b x^{3} + a^{6} x^{2}\right )} \sqrt{b x + a} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(5/2)*x^(7/2)),x, algorithm="fricas")

[Out]

-2/15*(3*A*a^4 - 16*(5*B*a*b^3 - 8*A*b^4)*x^4 - 24*(5*B*a^2*b^2 - 8*A*a*b^3)*x^3
- 6*(5*B*a^3*b - 8*A*a^2*b^2)*x^2 + (5*B*a^4 - 8*A*a^3*b)*x)/((a^5*b*x^3 + a^6*
x^2)*sqrt(b*x + a)*sqrt(x))

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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)/x**(7/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296747, size = 444, normalized size = 3.08 \[ -\frac{\sqrt{b x + a}{\left ({\left (b x + a\right )}{\left (\frac{{\left (40 \, B a^{8} b^{7} - 73 \, A a^{7} b^{8}\right )}{\left (b x + a\right )}}{a^{3} b^{9}} - \frac{5 \,{\left (17 \, B a^{9} b^{7} - 32 \, A a^{8} b^{8}\right )}}{a^{3} b^{9}}\right )} + \frac{45 \,{\left (B a^{10} b^{7} - 2 \, A a^{9} b^{8}\right )}}{a^{3} b^{9}}\right )}}{960 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{2}}} + \frac{4 \,{\left (6 \, B a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{7}{2}} + 18 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{9}{2}} - 9 \, A{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{9}{2}} + 8 \, B a^{3} b^{\frac{11}{2}} - 24 \, A a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{11}{2}} - 11 \, A a^{2} b^{\frac{13}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{4}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(5/2)*x^(7/2)),x, algorithm="giac")

[Out]

-1/960*sqrt(b*x + a)*((b*x + a)*((40*B*a^8*b^7 - 73*A*a^7*b^8)*(b*x + a)/(a^3*b^
9) - 5*(17*B*a^9*b^7 - 32*A*a^8*b^8)/(a^3*b^9)) + 45*(B*a^10*b^7 - 2*A*a^9*b^8)/
(a^3*b^9))/((b*x + a)*b - a*b)^(5/2) + 4/3*(6*B*a*(sqrt(b*x + a)*sqrt(b) - sqrt(
(b*x + a)*b - a*b))^4*b^(7/2) + 18*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)
*b - a*b))^2*b^(9/2) - 9*A*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b
^(9/2) + 8*B*a^3*b^(11/2) - 24*A*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a
*b))^2*b^(11/2) - 11*A*a^2*b^(13/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b
- a*b))^2 + a*b)^3*a^4*abs(b))