3.476 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^{13/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{32 b^3 (a+b x)^{3/2} (8 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac{16 b^2 (a+b x)^{3/2} (8 A b-11 a B)}{1155 a^4 x^{5/2}}-\frac{4 b (a+b x)^{3/2} (8 A b-11 a B)}{231 a^3 x^{7/2}}+\frac{2 (a+b x)^{3/2} (8 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{3/2}}{11 a x^{11/2}} \]

[Out]

(-2*A*(a + b*x)^(3/2))/(11*a*x^(11/2)) + (2*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(9
9*a^2*x^(9/2)) - (4*b*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(231*a^3*x^(7/2)) + (16*
b^2*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(1155*a^4*x^(5/2)) - (32*b^3*(8*A*b - 11*a
*B)*(a + b*x)^(3/2))/(3465*a^5*x^(3/2))

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Rubi [A]  time = 0.180673, antiderivative size = 150, 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 \[ -\frac{32 b^3 (a+b x)^{3/2} (8 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac{16 b^2 (a+b x)^{3/2} (8 A b-11 a B)}{1155 a^4 x^{5/2}}-\frac{4 b (a+b x)^{3/2} (8 A b-11 a B)}{231 a^3 x^{7/2}}+\frac{2 (a+b x)^{3/2} (8 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{3/2}}{11 a x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/x^(13/2),x]

[Out]

(-2*A*(a + b*x)^(3/2))/(11*a*x^(11/2)) + (2*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(9
9*a^2*x^(9/2)) - (4*b*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(231*a^3*x^(7/2)) + (16*
b^2*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(1155*a^4*x^(5/2)) - (32*b^3*(8*A*b - 11*a
*B)*(a + b*x)^(3/2))/(3465*a^5*x^(3/2))

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Rubi in Sympy [A]  time = 15.0572, size = 150, normalized size = 1. \[ - \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{11 a x^{\frac{11}{2}}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (8 A b - 11 B a\right )}{99 a^{2} x^{\frac{9}{2}}} - \frac{4 b \left (a + b x\right )^{\frac{3}{2}} \left (8 A b - 11 B a\right )}{231 a^{3} x^{\frac{7}{2}}} + \frac{16 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (8 A b - 11 B a\right )}{1155 a^{4} x^{\frac{5}{2}}} - \frac{32 b^{3} \left (a + b x\right )^{\frac{3}{2}} \left (8 A b - 11 B a\right )}{3465 a^{5} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/x**(13/2),x)

[Out]

-2*A*(a + b*x)**(3/2)/(11*a*x**(11/2)) + 2*(a + b*x)**(3/2)*(8*A*b - 11*B*a)/(99
*a**2*x**(9/2)) - 4*b*(a + b*x)**(3/2)*(8*A*b - 11*B*a)/(231*a**3*x**(7/2)) + 16
*b**2*(a + b*x)**(3/2)*(8*A*b - 11*B*a)/(1155*a**4*x**(5/2)) - 32*b**3*(a + b*x)
**(3/2)*(8*A*b - 11*B*a)/(3465*a**5*x**(3/2))

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Mathematica [A]  time = 0.103436, size = 95, normalized size = 0.63 \[ -\frac{2 (a+b x)^{3/2} \left (35 a^4 (9 A+11 B x)-10 a^3 b x (28 A+33 B x)+24 a^2 b^2 x^2 (10 A+11 B x)-16 a b^3 x^3 (12 A+11 B x)+128 A b^4 x^4\right )}{3465 a^5 x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(13/2),x]

[Out]

(-2*(a + b*x)^(3/2)*(128*A*b^4*x^4 + 35*a^4*(9*A + 11*B*x) + 24*a^2*b^2*x^2*(10*
A + 11*B*x) - 16*a*b^3*x^3*(12*A + 11*B*x) - 10*a^3*b*x*(28*A + 33*B*x)))/(3465*
a^5*x^(11/2))

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Maple [A]  time = 0.008, size = 101, normalized size = 0.7 \[ -{\frac{256\,A{b}^{4}{x}^{4}-352\,Ba{b}^{3}{x}^{4}-384\,Aa{b}^{3}{x}^{3}+528\,B{a}^{2}{b}^{2}{x}^{3}+480\,A{a}^{2}{b}^{2}{x}^{2}-660\,B{a}^{3}b{x}^{2}-560\,A{a}^{3}bx+770\,B{a}^{4}x+630\,A{a}^{4}}{3465\,{a}^{5}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/x^(13/2),x)

[Out]

-2/3465*(b*x+a)^(3/2)*(128*A*b^4*x^4-176*B*a*b^3*x^4-192*A*a*b^3*x^3+264*B*a^2*b
^2*x^3+240*A*a^2*b^2*x^2-330*B*a^3*b*x^2-280*A*a^3*b*x+385*B*a^4*x+315*A*a^4)/x^
(11/2)/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)*sqrt(b*x + a)/x^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234842, size = 169, normalized size = 1.13 \[ -\frac{2 \,{\left (315 \, A a^{5} - 16 \,{\left (11 \, B a b^{4} - 8 \, A b^{5}\right )} x^{5} + 8 \,{\left (11 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} x^{4} - 6 \,{\left (11 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} x^{3} + 5 \,{\left (11 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x^{2} + 35 \,{\left (11 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt{b x + a}}{3465 \, a^{5} x^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^(13/2),x, algorithm="fricas")

[Out]

-2/3465*(315*A*a^5 - 16*(11*B*a*b^4 - 8*A*b^5)*x^5 + 8*(11*B*a^2*b^3 - 8*A*a*b^4
)*x^4 - 6*(11*B*a^3*b^2 - 8*A*a^2*b^3)*x^3 + 5*(11*B*a^4*b - 8*A*a^3*b^2)*x^2 +
35*(11*B*a^5 + A*a^4*b)*x)*sqrt(b*x + a)/(a^5*x^(11/2))

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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**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232671, size = 248, normalized size = 1.65 \[ -\frac{{\left ({\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (11 \, B a b^{10} - 8 \, A b^{11}\right )}{\left (b x + a\right )}}{a^{6} b^{18}} - \frac{11 \,{\left (11 \, B a^{2} b^{10} - 8 \, A a b^{11}\right )}}{a^{6} b^{18}}\right )} + \frac{99 \,{\left (11 \, B a^{3} b^{10} - 8 \, A a^{2} b^{11}\right )}}{a^{6} b^{18}}\right )} - \frac{231 \,{\left (11 \, B a^{4} b^{10} - 8 \, A a^{3} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )} + \frac{1155 \,{\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )}^{\frac{3}{2}} b}{14192640 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{11}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^(13/2),x, algorithm="giac")

[Out]

-1/14192640*((2*(b*x + a)*(4*(b*x + a)*(2*(11*B*a*b^10 - 8*A*b^11)*(b*x + a)/(a^
6*b^18) - 11*(11*B*a^2*b^10 - 8*A*a*b^11)/(a^6*b^18)) + 99*(11*B*a^3*b^10 - 8*A*
a^2*b^11)/(a^6*b^18)) - 231*(11*B*a^4*b^10 - 8*A*a^3*b^11)/(a^6*b^18))*(b*x + a)
+ 1155*(B*a^5*b^10 - A*a^4*b^11)/(a^6*b^18))*(b*x + a)^(3/2)*b/(((b*x + a)*b -
a*b)^(11/2)*abs(b))