3.475 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^{11/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{16 b^2 (a+b x)^{3/2} (2 A b-3 a B)}{315 a^4 x^{3/2}}-\frac{8 b (a+b x)^{3/2} (2 A b-3 a B)}{105 a^3 x^{5/2}}+\frac{2 (a+b x)^{3/2} (2 A b-3 a B)}{21 a^2 x^{7/2}}-\frac{2 A (a+b x)^{3/2}}{9 a x^{9/2}} \]

[Out]

(-2*A*(a + b*x)^(3/2))/(9*a*x^(9/2)) + (2*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(21*a
^2*x^(7/2)) - (8*b*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(105*a^3*x^(5/2)) + (16*b^2*
(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(315*a^4*x^(3/2))

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Rubi [A]  time = 0.140575, 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 \[ \frac{16 b^2 (a+b x)^{3/2} (2 A b-3 a B)}{315 a^4 x^{3/2}}-\frac{8 b (a+b x)^{3/2} (2 A b-3 a B)}{105 a^3 x^{5/2}}+\frac{2 (a+b x)^{3/2} (2 A b-3 a B)}{21 a^2 x^{7/2}}-\frac{2 A (a+b x)^{3/2}}{9 a x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/x^(11/2),x]

[Out]

(-2*A*(a + b*x)^(3/2))/(9*a*x^(9/2)) + (2*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(21*a
^2*x^(7/2)) - (8*b*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(105*a^3*x^(5/2)) + (16*b^2*
(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(315*a^4*x^(3/2))

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Rubi in Sympy [A]  time = 11.3241, size = 116, normalized size = 0.99 \[ - \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{9 a x^{\frac{9}{2}}} + \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (A b - \frac{3 B a}{2}\right )}{21 a^{2} x^{\frac{7}{2}}} - \frac{8 b \left (a + b x\right )^{\frac{3}{2}} \left (2 A b - 3 B a\right )}{105 a^{3} x^{\frac{5}{2}}} + \frac{32 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (A b - \frac{3 B a}{2}\right )}{315 a^{4} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/x**(11/2),x)

[Out]

-2*A*(a + b*x)**(3/2)/(9*a*x**(9/2)) + 4*(a + b*x)**(3/2)*(A*b - 3*B*a/2)/(21*a*
*2*x**(7/2)) - 8*b*(a + b*x)**(3/2)*(2*A*b - 3*B*a)/(105*a**3*x**(5/2)) + 32*b**
2*(a + b*x)**(3/2)*(A*b - 3*B*a/2)/(315*a**4*x**(3/2))

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Mathematica [A]  time = 0.0808287, size = 73, normalized size = 0.62 \[ -\frac{2 (a+b x)^{3/2} \left (5 a^3 (7 A+9 B x)-6 a^2 b x (5 A+6 B x)+24 a b^2 x^2 (A+B x)-16 A b^3 x^3\right )}{315 a^4 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(11/2),x]

[Out]

(-2*(a + b*x)^(3/2)*(-16*A*b^3*x^3 + 24*a*b^2*x^2*(A + B*x) - 6*a^2*b*x*(5*A + 6
*B*x) + 5*a^3*(7*A + 9*B*x)))/(315*a^4*x^(9/2))

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Maple [A]  time = 0.008, size = 77, normalized size = 0.7 \[ -{\frac{-32\,A{b}^{3}{x}^{3}+48\,B{x}^{3}a{b}^{2}+48\,aA{b}^{2}{x}^{2}-72\,B{x}^{2}{a}^{2}b-60\,{a}^{2}Abx+90\,{a}^{3}Bx+70\,A{a}^{3}}{315\,{a}^{4}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/x^(11/2),x)

[Out]

-2/315*(b*x+a)^(3/2)*(-16*A*b^3*x^3+24*B*a*b^2*x^3+24*A*a*b^2*x^2-36*B*a^2*b*x^2
-30*A*a^2*b*x+45*B*a^3*x+35*A*a^3)/x^(9/2)/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^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228513, size = 136, normalized size = 1.16 \[ -\frac{2 \,{\left (35 \, A a^{4} + 8 \,{\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} - 4 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 3 \,{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 5 \,{\left (9 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt{b x + a}}{315 \, a^{4} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^(11/2),x, algorithm="fricas")

[Out]

-2/315*(35*A*a^4 + 8*(3*B*a*b^3 - 2*A*b^4)*x^4 - 4*(3*B*a^2*b^2 - 2*A*a*b^3)*x^3
+ 3*(3*B*a^3*b - 2*A*a^2*b^2)*x^2 + 5*(9*B*a^4 + A*a^3*b)*x)*sqrt(b*x + a)/(a^4
*x^(9/2))

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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)*(b*x+a)**(1/2)/x**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227707, size = 201, normalized size = 1.72 \[ \frac{{\left ({\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (3 \, B a b^{8} - 2 \, A b^{9}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (3 \, B a^{2} b^{8} - 2 \, A a b^{9}\right )}}{a^{5} b^{15}}\right )} + \frac{63 \,{\left (3 \, B a^{3} b^{8} - 2 \, A a^{2} b^{9}\right )}}{a^{5} b^{15}}\right )} - \frac{105 \,{\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{5} b^{15}}\right )}{\left (b x + a\right )}^{\frac{3}{2}} b}{322560 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{9}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^(11/2),x, algorithm="giac")

[Out]

1/322560*((b*x + a)*(4*(b*x + a)*(2*(3*B*a*b^8 - 2*A*b^9)*(b*x + a)/(a^5*b^15) -
9*(3*B*a^2*b^8 - 2*A*a*b^9)/(a^5*b^15)) + 63*(3*B*a^3*b^8 - 2*A*a^2*b^9)/(a^5*b
^15)) - 105*(B*a^4*b^8 - A*a^3*b^9)/(a^5*b^15))*(b*x + a)^(3/2)*b/(((b*x + a)*b
- a*b)^(9/2)*abs(b))