3.510 \(\int \frac{A+B x}{x^{9/2} \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=117 \[ \frac{16 b^2 \sqrt{a+b x} (6 A b-7 a B)}{105 a^4 \sqrt{x}}-\frac{8 b \sqrt{a+b x} (6 A b-7 a B)}{105 a^3 x^{3/2}}+\frac{2 \sqrt{a+b x} (6 A b-7 a B)}{35 a^2 x^{5/2}}-\frac{2 A \sqrt{a+b x}}{7 a x^{7/2}} \]

[Out]

(-2*A*Sqrt[a + b*x])/(7*a*x^(7/2)) + (2*(6*A*b - 7*a*B)*Sqrt[a + b*x])/(35*a^2*x
^(5/2)) - (8*b*(6*A*b - 7*a*B)*Sqrt[a + b*x])/(105*a^3*x^(3/2)) + (16*b^2*(6*A*b
- 7*a*B)*Sqrt[a + b*x])/(105*a^4*Sqrt[x])

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Rubi [A]  time = 0.139437, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{16 b^2 \sqrt{a+b x} (6 A b-7 a B)}{105 a^4 \sqrt{x}}-\frac{8 b \sqrt{a+b x} (6 A b-7 a B)}{105 a^3 x^{3/2}}+\frac{2 \sqrt{a+b x} (6 A b-7 a B)}{35 a^2 x^{5/2}}-\frac{2 A \sqrt{a+b x}}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(9/2)*Sqrt[a + b*x]),x]

[Out]

(-2*A*Sqrt[a + b*x])/(7*a*x^(7/2)) + (2*(6*A*b - 7*a*B)*Sqrt[a + b*x])/(35*a^2*x
^(5/2)) - (8*b*(6*A*b - 7*a*B)*Sqrt[a + b*x])/(105*a^3*x^(3/2)) + (16*b^2*(6*A*b
- 7*a*B)*Sqrt[a + b*x])/(105*a^4*Sqrt[x])

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Rubi in Sympy [A]  time = 11.125, size = 116, normalized size = 0.99 \[ - \frac{2 A \sqrt{a + b x}}{7 a x^{\frac{7}{2}}} + \frac{2 \sqrt{a + b x} \left (6 A b - 7 B a\right )}{35 a^{2} x^{\frac{5}{2}}} - \frac{8 b \sqrt{a + b x} \left (6 A b - 7 B a\right )}{105 a^{3} x^{\frac{3}{2}}} + \frac{16 b^{2} \sqrt{a + b x} \left (6 A b - 7 B a\right )}{105 a^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(9/2)/(b*x+a)**(1/2),x)

[Out]

-2*A*sqrt(a + b*x)/(7*a*x**(7/2)) + 2*sqrt(a + b*x)*(6*A*b - 7*B*a)/(35*a**2*x**
(5/2)) - 8*b*sqrt(a + b*x)*(6*A*b - 7*B*a)/(105*a**3*x**(3/2)) + 16*b**2*sqrt(a
+ b*x)*(6*A*b - 7*B*a)/(105*a**4*sqrt(x))

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Mathematica [A]  time = 0.0782835, size = 76, normalized size = 0.65 \[ -\frac{2 \sqrt{a+b x} \left (3 a^3 (5 A+7 B x)-2 a^2 b x (9 A+14 B x)+8 a b^2 x^2 (3 A+7 B x)-48 A b^3 x^3\right )}{105 a^4 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(9/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x]*(-48*A*b^3*x^3 + 8*a*b^2*x^2*(3*A + 7*B*x) + 3*a^3*(5*A + 7*B*
x) - 2*a^2*b*x*(9*A + 14*B*x)))/(105*a^4*x^(7/2))

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Maple [A]  time = 0.007, size = 77, normalized size = 0.7 \[ -{\frac{-96\,A{b}^{3}{x}^{3}+112\,B{x}^{3}a{b}^{2}+48\,aA{b}^{2}{x}^{2}-56\,B{x}^{2}{a}^{2}b-36\,{a}^{2}Abx+42\,{a}^{3}Bx+30\,A{a}^{3}}{105\,{a}^{4}}\sqrt{bx+a}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(9/2)/(b*x+a)^(1/2),x)

[Out]

-2/105*(b*x+a)^(1/2)*(-48*A*b^3*x^3+56*B*a*b^2*x^3+24*A*a*b^2*x^2-28*B*a^2*b*x^2
-18*A*a^2*b*x+21*B*a^3*x+15*A*a^3)/x^(7/2)/a^4

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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)/(sqrt(b*x + a)*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23389, size = 105, normalized size = 0.9 \[ -\frac{2 \,{\left (15 \, A a^{3} + 8 \,{\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} - 4 \,{\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{2} + 3 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} \sqrt{b x + a}}{105 \, a^{4} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*x^(9/2)),x, algorithm="fricas")

[Out]

-2/105*(15*A*a^3 + 8*(7*B*a*b^2 - 6*A*b^3)*x^3 - 4*(7*B*a^2*b - 6*A*a*b^2)*x^2 +
3*(7*B*a^3 - 6*A*a^2*b)*x)*sqrt(b*x + a)/(a^4*x^(7/2))

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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**(9/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224799, size = 201, normalized size = 1.72 \[ \frac{{\left ({\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (7 \, B a b^{6} - 6 \, A b^{7}\right )}{\left (b x + a\right )}}{a^{4} b^{12}} - \frac{7 \,{\left (7 \, B a^{2} b^{6} - 6 \, A a b^{7}\right )}}{a^{4} b^{12}}\right )} + \frac{35 \,{\left (7 \, B a^{3} b^{6} - 6 \, A a^{2} b^{7}\right )}}{a^{4} b^{12}}\right )} - \frac{105 \,{\left (B a^{4} b^{6} - A a^{3} b^{7}\right )}}{a^{4} b^{12}}\right )} \sqrt{b x + a} b}{80640 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*x^(9/2)),x, algorithm="giac")

[Out]

1/80640*((b*x + a)*(4*(b*x + a)*(2*(7*B*a*b^6 - 6*A*b^7)*(b*x + a)/(a^4*b^12) -
7*(7*B*a^2*b^6 - 6*A*a*b^7)/(a^4*b^12)) + 35*(7*B*a^3*b^6 - 6*A*a^2*b^7)/(a^4*b^
12)) - 105*(B*a^4*b^6 - A*a^3*b^7)/(a^4*b^12))*sqrt(b*x + a)*b/(((b*x + a)*b - a
*b)^(7/2)*abs(b))