3.832 \(\int \frac{A+B x}{\sqrt{x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=258 \[ \frac{\sqrt{x} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (a B+7 A b)}{24 a^2 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (a+b x) (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{9/2} b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a B+7 A b)}{64 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a B+7 A b)}{96 a^3 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(5*(7*A*b + a*B)*Sqrt[x])/(64*a^4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B
)*Sqrt[x])/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((7*A*b + a*B)*Sq
rt[x])/(24*a^2*b*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(7*A*b + a*B)*S
qrt[x])/(96*a^3*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(7*A*b + a*B)*(a
+ b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(9/2)*b^(3/2)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])

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Rubi [A]  time = 0.36881, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{\sqrt{x} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (a B+7 A b)}{24 a^2 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (a+b x) (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{9/2} b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a B+7 A b)}{64 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a B+7 A b)}{96 a^3 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(5*(7*A*b + a*B)*Sqrt[x])/(64*a^4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B
)*Sqrt[x])/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((7*A*b + a*B)*Sq
rt[x])/(24*a^2*b*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(7*A*b + a*B)*S
qrt[x])/(96*a^3*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(7*A*b + a*B)*(a
+ b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(9/2)*b^(3/2)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

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)**(5/2)/x**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.135006, size = 179, normalized size = 0.69 \[ \frac{-\frac{48 a^{7/2} \sqrt{b} \sqrt{x} (a B-A b)}{a+b x}+8 a^{5/2} \sqrt{b} \sqrt{x} (a B+7 A b)+10 a^{3/2} \sqrt{b} \sqrt{x} (a+b x) (a B+7 A b)+15 \sqrt{a} \sqrt{b} \sqrt{x} (a+b x)^2 (a B+7 A b)+15 (a+b x)^3 (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{192 a^{9/2} b^{3/2} \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(8*a^(5/2)*Sqrt[b]*(7*A*b + a*B)*Sqrt[x] - (48*a^(7/2)*Sqrt[b]*(-(A*b) + a*B)*Sq
rt[x])/(a + b*x) + 10*a^(3/2)*Sqrt[b]*(7*A*b + a*B)*Sqrt[x]*(a + b*x) + 15*Sqrt[
a]*Sqrt[b]*(7*A*b + a*B)*Sqrt[x]*(a + b*x)^2 + 15*(7*A*b + a*B)*(a + b*x)^3*ArcT
an[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(192*a^(9/2)*b^(3/2)*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.026, size = 357, normalized size = 1.4 \[{\frac{bx+a}{192\,{a}^{4}b} \left ( 105\,A\sqrt{ab}{x}^{7/2}{b}^{4}+15\,B\sqrt{ab}{x}^{7/2}a{b}^{3}+385\,A\sqrt{ab}{x}^{5/2}a{b}^{3}+105\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}{b}^{5}+55\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}+15\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}a{b}^{4}+420\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}a{b}^{4}+60\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{3}+511\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}+630\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{3}+73\,B\sqrt{ab}{x}^{3/2}{a}^{3}b+90\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{2}+420\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{3}{b}^{2}+60\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{4}b+279\,A\sqrt{ab}\sqrt{x}{a}^{3}b+105\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4}b-15\,B\sqrt{ab}\sqrt{x}{a}^{4}+15\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2)/x^(1/2),x)

[Out]

1/192*(105*A*(a*b)^(1/2)*x^(7/2)*b^4+15*B*(a*b)^(1/2)*x^(7/2)*a*b^3+385*A*(a*b)^
(1/2)*x^(5/2)*a*b^3+105*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^4*b^5+55*B*(a*b)^(1/2)
*x^(5/2)*a^2*b^2+15*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^4*a*b^4+420*A*arctan(x^(1/
2)*b/(a*b)^(1/2))*x^3*a*b^4+60*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^3*a^2*b^3+511*A
*(a*b)^(1/2)*x^(3/2)*a^2*b^2+630*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^2*b^3+73*
B*(a*b)^(1/2)*x^(3/2)*a^3*b+90*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^3*b^2+420*A
*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^3*b^2+60*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^
4*b+279*A*(a*b)^(1/2)*x^(1/2)*a^3*b+105*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a^4*b-15
*B*(a*b)^(1/2)*x^(1/2)*a^4+15*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^5)*(b*x+a)/(a*b)
^(1/2)/b/a^4/((b*x+a)^2)^(5/2)

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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)^(5/2)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296148, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (15 \, B a^{4} - 279 \, A a^{3} b - 15 \,{\left (B a b^{3} + 7 \, A b^{4}\right )} x^{3} - 55 \,{\left (B a^{2} b^{2} + 7 \, A a b^{3}\right )} x^{2} - 73 \,{\left (B a^{3} b + 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 15 \,{\left (B a^{5} + 7 \, A a^{4} b +{\left (B a b^{4} + 7 \, A b^{5}\right )} x^{4} + 4 \,{\left (B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} + 6 \,{\left (B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (B a^{4} b + 7 \, A a^{3} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{384 \,{\left (a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{3} + 6 \, a^{6} b^{3} x^{2} + 4 \, a^{7} b^{2} x + a^{8} b\right )} \sqrt{-a b}}, -\frac{{\left (15 \, B a^{4} - 279 \, A a^{3} b - 15 \,{\left (B a b^{3} + 7 \, A b^{4}\right )} x^{3} - 55 \,{\left (B a^{2} b^{2} + 7 \, A a b^{3}\right )} x^{2} - 73 \,{\left (B a^{3} b + 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 15 \,{\left (B a^{5} + 7 \, A a^{4} b +{\left (B a b^{4} + 7 \, A b^{5}\right )} x^{4} + 4 \,{\left (B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} + 6 \,{\left (B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (B a^{4} b + 7 \, A a^{3} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{192 \,{\left (a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{3} + 6 \, a^{6} b^{3} x^{2} + 4 \, a^{7} b^{2} x + a^{8} b\right )} \sqrt{a b}}\right ] \]

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)^(5/2)*sqrt(x)),x, algorithm="fricas")

[Out]

[-1/384*(2*(15*B*a^4 - 279*A*a^3*b - 15*(B*a*b^3 + 7*A*b^4)*x^3 - 55*(B*a^2*b^2
+ 7*A*a*b^3)*x^2 - 73*(B*a^3*b + 7*A*a^2*b^2)*x)*sqrt(-a*b)*sqrt(x) - 15*(B*a^5
+ 7*A*a^4*b + (B*a*b^4 + 7*A*b^5)*x^4 + 4*(B*a^2*b^3 + 7*A*a*b^4)*x^3 + 6*(B*a^3
*b^2 + 7*A*a^2*b^3)*x^2 + 4*(B*a^4*b + 7*A*a^3*b^2)*x)*log((2*a*b*sqrt(x) + sqrt
(-a*b)*(b*x - a))/(b*x + a)))/((a^4*b^5*x^4 + 4*a^5*b^4*x^3 + 6*a^6*b^3*x^2 + 4*
a^7*b^2*x + a^8*b)*sqrt(-a*b)), -1/192*((15*B*a^4 - 279*A*a^3*b - 15*(B*a*b^3 +
7*A*b^4)*x^3 - 55*(B*a^2*b^2 + 7*A*a*b^3)*x^2 - 73*(B*a^3*b + 7*A*a^2*b^2)*x)*sq
rt(a*b)*sqrt(x) + 15*(B*a^5 + 7*A*a^4*b + (B*a*b^4 + 7*A*b^5)*x^4 + 4*(B*a^2*b^3
+ 7*A*a*b^4)*x^3 + 6*(B*a^3*b^2 + 7*A*a^2*b^3)*x^2 + 4*(B*a^4*b + 7*A*a^3*b^2)*
x)*arctan(a/(sqrt(a*b)*sqrt(x))))/((a^4*b^5*x^4 + 4*a^5*b^4*x^3 + 6*a^6*b^3*x^2
+ 4*a^7*b^2*x + a^8*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)**(5/2)/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278191, size = 198, normalized size = 0.77 \[ \frac{5 \,{\left (B a + 7 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} a^{4} b{\rm sign}\left (b x + a\right )} + \frac{15 \, B a b^{3} x^{\frac{7}{2}} + 105 \, A b^{4} x^{\frac{7}{2}} + 55 \, B a^{2} b^{2} x^{\frac{5}{2}} + 385 \, A a b^{3} x^{\frac{5}{2}} + 73 \, B a^{3} b x^{\frac{3}{2}} + 511 \, A a^{2} b^{2} x^{\frac{3}{2}} - 15 \, B a^{4} \sqrt{x} + 279 \, A a^{3} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} a^{4} b{\rm sign}\left (b x + a\right )} \]

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)^(5/2)*sqrt(x)),x, algorithm="giac")

[Out]

5/64*(B*a + 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4*b*sign(b*x + a)) +
1/192*(15*B*a*b^3*x^(7/2) + 105*A*b^4*x^(7/2) + 55*B*a^2*b^2*x^(5/2) + 385*A*a*
b^3*x^(5/2) + 73*B*a^3*b*x^(3/2) + 511*A*a^2*b^2*x^(3/2) - 15*B*a^4*sqrt(x) + 27
9*A*a^3*b*sqrt(x))/((b*x + a)^4*a^4*b*sign(b*x + a))