3.779 \(\int \frac{A+B x}{\sqrt{x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=190 \[ \frac{7 (a B+9 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{11/2} b^{3/2}}+\frac{7 \sqrt{x} (a B+9 A b)}{128 a^5 b (a+b x)}+\frac{7 \sqrt{x} (a B+9 A b)}{192 a^4 b (a+b x)^2}+\frac{7 \sqrt{x} (a B+9 A b)}{240 a^3 b (a+b x)^3}+\frac{\sqrt{x} (a B+9 A b)}{40 a^2 b (a+b x)^4}+\frac{\sqrt{x} (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

((A*b - a*B)*Sqrt[x])/(5*a*b*(a + b*x)^5) + ((9*A*b + a*B)*Sqrt[x])/(40*a^2*b*(a
+ b*x)^4) + (7*(9*A*b + a*B)*Sqrt[x])/(240*a^3*b*(a + b*x)^3) + (7*(9*A*b + a*B
)*Sqrt[x])/(192*a^4*b*(a + b*x)^2) + (7*(9*A*b + a*B)*Sqrt[x])/(128*a^5*b*(a + b
*x)) + (7*(9*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(11/2)*b^(3/2)
)

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Rubi [A]  time = 0.213766, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{7 (a B+9 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{11/2} b^{3/2}}+\frac{7 \sqrt{x} (a B+9 A b)}{128 a^5 b (a+b x)}+\frac{7 \sqrt{x} (a B+9 A b)}{192 a^4 b (a+b x)^2}+\frac{7 \sqrt{x} (a B+9 A b)}{240 a^3 b (a+b x)^3}+\frac{\sqrt{x} (a B+9 A b)}{40 a^2 b (a+b x)^4}+\frac{\sqrt{x} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(Sqrt[x]*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((A*b - a*B)*Sqrt[x])/(5*a*b*(a + b*x)^5) + ((9*A*b + a*B)*Sqrt[x])/(40*a^2*b*(a
+ b*x)^4) + (7*(9*A*b + a*B)*Sqrt[x])/(240*a^3*b*(a + b*x)^3) + (7*(9*A*b + a*B
)*Sqrt[x])/(192*a^4*b*(a + b*x)^2) + (7*(9*A*b + a*B)*Sqrt[x])/(128*a^5*b*(a + b
*x)) + (7*(9*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(11/2)*b^(3/2)
)

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Rubi in Sympy [A]  time = 58.6843, size = 173, normalized size = 0.91 \[ \frac{\sqrt{x} \left (A b - B a\right )}{5 a b \left (a + b x\right )^{5}} + \frac{\sqrt{x} \left (9 A b + B a\right )}{40 a^{2} b \left (a + b x\right )^{4}} + \frac{7 \sqrt{x} \left (9 A b + B a\right )}{240 a^{3} b \left (a + b x\right )^{3}} + \frac{7 \sqrt{x} \left (9 A b + B a\right )}{192 a^{4} b \left (a + b x\right )^{2}} + \frac{7 \sqrt{x} \left (9 A b + B a\right )}{128 a^{5} b \left (a + b x\right )} + \frac{7 \left (9 A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 a^{\frac{11}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3/x**(1/2),x)

[Out]

sqrt(x)*(A*b - B*a)/(5*a*b*(a + b*x)**5) + sqrt(x)*(9*A*b + B*a)/(40*a**2*b*(a +
b*x)**4) + 7*sqrt(x)*(9*A*b + B*a)/(240*a**3*b*(a + b*x)**3) + 7*sqrt(x)*(9*A*b
+ B*a)/(192*a**4*b*(a + b*x)**2) + 7*sqrt(x)*(9*A*b + B*a)/(128*a**5*b*(a + b*x
)) + 7*(9*A*b + B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(128*a**(11/2)*b**(3/2))

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Mathematica [A]  time = 0.294923, size = 144, normalized size = 0.76 \[ \frac{7 (a B+9 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{11/2} b^{3/2}}+\frac{\sqrt{x} \left (-105 a^5 B+5 a^4 b (579 A+158 B x)+2 a^3 b^2 x (3555 A+448 B x)+14 a^2 b^3 x^2 (576 A+35 B x)+105 a b^4 x^3 (42 A+B x)+945 A b^5 x^4\right )}{1920 a^5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(Sqrt[x]*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(Sqrt[x]*(-105*a^5*B + 945*A*b^5*x^4 + 105*a*b^4*x^3*(42*A + B*x) + 14*a^2*b^3*x
^2*(576*A + 35*B*x) + 5*a^4*b*(579*A + 158*B*x) + 2*a^3*b^2*x*(3555*A + 448*B*x)
))/(1920*a^5*b*(a + b*x)^5) + (7*(9*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]
)/(128*a^(11/2)*b^(3/2))

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Maple [A]  time = 0.027, size = 150, normalized size = 0.8 \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{5}} \left ({\frac{ \left ( 63\,Ab+7\,Ba \right ){b}^{3}{x}^{9/2}}{256\,{a}^{5}}}+{\frac{49\,{b}^{2} \left ( 9\,Ab+Ba \right ){x}^{7/2}}{384\,{a}^{4}}}+{\frac{ \left ( 63\,Ab+7\,Ba \right ) b{x}^{5/2}}{30\,{a}^{3}}}+{\frac{ \left ( 711\,Ab+79\,Ba \right ){x}^{3/2}}{384\,{a}^{2}}}+{\frac{ \left ( 193\,Ab-7\,Ba \right ) \sqrt{x}}{256\,ab}} \right ) }+{\frac{63\,A}{128\,{a}^{5}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{7\,B}{128\,{a}^{4}b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3/x^(1/2),x)

[Out]

2*(7/256*(9*A*b+B*a)/a^5*b^3*x^(9/2)+49/384/a^4*b^2*(9*A*b+B*a)*x^(7/2)+7/30/a^3
*(9*A*b+B*a)*b*x^(5/2)+79/384/a^2*(9*A*b+B*a)*x^(3/2)+1/256*(193*A*b-7*B*a)/a/b*
x^(1/2))/(b*x+a)^5+63/128/a^5/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+7/128/
a^4/b/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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^2*x^2 + 2*a*b*x + a^2)^3*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.324633, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (105 \, B a^{5} - 2895 \, A a^{4} b - 105 \,{\left (B a b^{4} + 9 \, A b^{5}\right )} x^{4} - 490 \,{\left (B a^{2} b^{3} + 9 \, A a b^{4}\right )} x^{3} - 896 \,{\left (B a^{3} b^{2} + 9 \, A a^{2} b^{3}\right )} x^{2} - 790 \,{\left (B a^{4} b + 9 \, A a^{3} b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 105 \,{\left (B a^{6} + 9 \, A a^{5} b +{\left (B a b^{5} + 9 \, A b^{6}\right )} x^{5} + 5 \,{\left (B a^{2} b^{4} + 9 \, A a b^{5}\right )} x^{4} + 10 \,{\left (B a^{3} b^{3} + 9 \, A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (B a^{4} b^{2} + 9 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (B a^{5} b + 9 \, A a^{4} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{3840 \,{\left (a^{5} b^{6} x^{5} + 5 \, a^{6} b^{5} x^{4} + 10 \, a^{7} b^{4} x^{3} + 10 \, a^{8} b^{3} x^{2} + 5 \, a^{9} b^{2} x + a^{10} b\right )} \sqrt{-a b}}, -\frac{{\left (105 \, B a^{5} - 2895 \, A a^{4} b - 105 \,{\left (B a b^{4} + 9 \, A b^{5}\right )} x^{4} - 490 \,{\left (B a^{2} b^{3} + 9 \, A a b^{4}\right )} x^{3} - 896 \,{\left (B a^{3} b^{2} + 9 \, A a^{2} b^{3}\right )} x^{2} - 790 \,{\left (B a^{4} b + 9 \, A a^{3} b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 105 \,{\left (B a^{6} + 9 \, A a^{5} b +{\left (B a b^{5} + 9 \, A b^{6}\right )} x^{5} + 5 \,{\left (B a^{2} b^{4} + 9 \, A a b^{5}\right )} x^{4} + 10 \,{\left (B a^{3} b^{3} + 9 \, A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (B a^{4} b^{2} + 9 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (B a^{5} b + 9 \, A a^{4} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{1920 \,{\left (a^{5} b^{6} x^{5} + 5 \, a^{6} b^{5} x^{4} + 10 \, a^{7} b^{4} x^{3} + 10 \, a^{8} b^{3} x^{2} + 5 \, a^{9} b^{2} x + a^{10} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^3*sqrt(x)),x, algorithm="fricas")

[Out]

[-1/3840*(2*(105*B*a^5 - 2895*A*a^4*b - 105*(B*a*b^4 + 9*A*b^5)*x^4 - 490*(B*a^2
*b^3 + 9*A*a*b^4)*x^3 - 896*(B*a^3*b^2 + 9*A*a^2*b^3)*x^2 - 790*(B*a^4*b + 9*A*a
^3*b^2)*x)*sqrt(-a*b)*sqrt(x) - 105*(B*a^6 + 9*A*a^5*b + (B*a*b^5 + 9*A*b^6)*x^5
+ 5*(B*a^2*b^4 + 9*A*a*b^5)*x^4 + 10*(B*a^3*b^3 + 9*A*a^2*b^4)*x^3 + 10*(B*a^4*
b^2 + 9*A*a^3*b^3)*x^2 + 5*(B*a^5*b + 9*A*a^4*b^2)*x)*log((2*a*b*sqrt(x) + sqrt(
-a*b)*(b*x - a))/(b*x + a)))/((a^5*b^6*x^5 + 5*a^6*b^5*x^4 + 10*a^7*b^4*x^3 + 10
*a^8*b^3*x^2 + 5*a^9*b^2*x + a^10*b)*sqrt(-a*b)), -1/1920*((105*B*a^5 - 2895*A*a
^4*b - 105*(B*a*b^4 + 9*A*b^5)*x^4 - 490*(B*a^2*b^3 + 9*A*a*b^4)*x^3 - 896*(B*a^
3*b^2 + 9*A*a^2*b^3)*x^2 - 790*(B*a^4*b + 9*A*a^3*b^2)*x)*sqrt(a*b)*sqrt(x) + 10
5*(B*a^6 + 9*A*a^5*b + (B*a*b^5 + 9*A*b^6)*x^5 + 5*(B*a^2*b^4 + 9*A*a*b^5)*x^4 +
10*(B*a^3*b^3 + 9*A*a^2*b^4)*x^3 + 10*(B*a^4*b^2 + 9*A*a^3*b^3)*x^2 + 5*(B*a^5*
b + 9*A*a^4*b^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x))))/((a^5*b^6*x^5 + 5*a^6*b^5*x^4
+ 10*a^7*b^4*x^3 + 10*a^8*b^3*x^2 + 5*a^9*b^2*x + a^10*b)*sqrt(a*b))]

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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**2*x**2+2*a*b*x+a**2)**3/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273833, size = 209, normalized size = 1.1 \[ \frac{7 \,{\left (B a + 9 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} a^{5} b} + \frac{105 \, B a b^{4} x^{\frac{9}{2}} + 945 \, A b^{5} x^{\frac{9}{2}} + 490 \, B a^{2} b^{3} x^{\frac{7}{2}} + 4410 \, A a b^{4} x^{\frac{7}{2}} + 896 \, B a^{3} b^{2} x^{\frac{5}{2}} + 8064 \, A a^{2} b^{3} x^{\frac{5}{2}} + 790 \, B a^{4} b x^{\frac{3}{2}} + 7110 \, A a^{3} b^{2} x^{\frac{3}{2}} - 105 \, B a^{5} \sqrt{x} + 2895 \, A a^{4} b \sqrt{x}}{1920 \,{\left (b x + a\right )}^{5} a^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^3*sqrt(x)),x, algorithm="giac")

[Out]

7/128*(B*a + 9*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5*b) + 1/1920*(105*
B*a*b^4*x^(9/2) + 945*A*b^5*x^(9/2) + 490*B*a^2*b^3*x^(7/2) + 4410*A*a*b^4*x^(7/
2) + 896*B*a^3*b^2*x^(5/2) + 8064*A*a^2*b^3*x^(5/2) + 790*B*a^4*b*x^(3/2) + 7110
*A*a^3*b^2*x^(3/2) - 105*B*a^5*sqrt(x) + 2895*A*a^4*b*sqrt(x))/((b*x + a)^5*a^5*
b)