3.835 \(\int \frac{A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=404 \[ \frac{13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 b^{3/2} (a+b x) (13 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 b (a+b x) (13 A b-5 a B)}{64 a^7 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 (a+b x) (13 A b-5 a B)}{64 a^6 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 (a+b x) (13 A b-5 a B)}{320 a^5 b x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(33*(13*A*b - 5*a*B))/(64*a^4*b*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b -
a*B)/(4*a*b*x^(5/2)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (13*A*b - 5*a*B
)/(24*a^2*b*x^(5/2)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*(13*A*b - 5
*a*B))/(96*a^3*b*x^(5/2)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*(13*A*b
- 5*a*B)*(a + b*x))/(320*a^5*b*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (77*(13
*A*b - 5*a*B)*(a + b*x))/(64*a^6*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*b
*(13*A*b - 5*a*B)*(a + b*x))/(64*a^7*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2
31*b^(3/2)*(13*A*b - 5*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(
15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.535327, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 b^{3/2} (a+b x) (13 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 b (a+b x) (13 A b-5 a B)}{64 a^7 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 (a+b x) (13 A b-5 a B)}{64 a^6 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 (a+b x) (13 A b-5 a B)}{320 a^5 b x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(33*(13*A*b - 5*a*B))/(64*a^4*b*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b -
a*B)/(4*a*b*x^(5/2)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (13*A*b - 5*a*B
)/(24*a^2*b*x^(5/2)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*(13*A*b - 5
*a*B))/(96*a^3*b*x^(5/2)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*(13*A*b
- 5*a*B)*(a + b*x))/(320*a^5*b*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (77*(13
*A*b - 5*a*B)*(a + b*x))/(64*a^6*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*b
*(13*A*b - 5*a*B)*(a + b*x))/(64*a^7*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2
31*b^(3/2)*(13*A*b - 5*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(
15/2)*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)/x**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.248517, size = 196, normalized size = 0.49 \[ \frac{\sqrt{a} \left (-128 a^6 (3 A+5 B x)+128 a^5 b x (13 A+55 B x)+11 a^4 b^2 x^2 (4185 B x-1664 A)+33 a^3 b^3 x^3 (2555 B x-3627 A)+231 a^2 b^4 x^4 (275 B x-949 A)+1155 a b^5 x^5 (15 B x-143 A)-45045 A b^6 x^6\right )+3465 b^{3/2} x^{5/2} (a+b x)^4 (5 a B-13 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{960 a^{15/2} x^{5/2} (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(Sqrt[a]*(-45045*A*b^6*x^6 - 128*a^6*(3*A + 5*B*x) + 1155*a*b^5*x^5*(-143*A + 15
*B*x) + 128*a^5*b*x*(13*A + 55*B*x) + 231*a^2*b^4*x^4*(-949*A + 275*B*x) + 33*a^
3*b^3*x^3*(-3627*A + 2555*B*x) + 11*a^4*b^2*x^2*(-1664*A + 4185*B*x)) + 3465*b^(
3/2)*(-13*A*b + 5*a*B)*x^(5/2)*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(9
60*a^(15/2)*x^(5/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.038, size = 449, normalized size = 1.1 \[ -{\frac{bx+a}{960\,{a}^{7}} \left ( -69300\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}{a}^{4}{b}^{3}-69300\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{11/2}{a}^{2}{b}^{5}+270270\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{9/2}{a}^{2}{b}^{5}-103950\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{9/2}{a}^{3}{b}^{4}+180180\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}{a}^{3}{b}^{4}-17325\,B\sqrt{ab}{x}^{6}a{b}^{5}+165165\,A\sqrt{ab}{x}^{5}a{b}^{5}+45045\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{13/2}{b}^{7}-63525\,B\sqrt{ab}{x}^{5}{a}^{2}{b}^{4}+219219\,A\sqrt{ab}{x}^{4}{a}^{2}{b}^{4}-84315\,B\sqrt{ab}{x}^{4}{a}^{3}{b}^{3}+119691\,A\sqrt{ab}{x}^{3}{a}^{3}{b}^{3}+45045\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}{a}^{4}{b}^{3}-17325\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{13/2}a{b}^{6}+180180\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{11/2}a{b}^{6}-46035\,B\sqrt{ab}{x}^{3}{a}^{4}{b}^{2}-17325\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}{a}^{5}{b}^{2}+18304\,A\sqrt{ab}{x}^{2}{a}^{4}{b}^{2}-7040\,B\sqrt{ab}{x}^{2}{a}^{5}b-1664\,A\sqrt{ab}x{a}^{5}b+45045\,A\sqrt{ab}{x}^{6}{b}^{6}+640\,B\sqrt{ab}x{a}^{6}+384\,A\sqrt{ab}{a}^{6} \right ){\frac{1}{\sqrt{ab}}}{x}^{-{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/960*(-69300*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(7/2)*a^4*b^3-69300*B*arctan(x^
(1/2)*b/(a*b)^(1/2))*x^(11/2)*a^2*b^5+270270*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(
9/2)*a^2*b^5-103950*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(9/2)*a^3*b^4+180180*A*arc
tan(x^(1/2)*b/(a*b)^(1/2))*x^(7/2)*a^3*b^4-17325*B*(a*b)^(1/2)*x^6*a*b^5+165165*
A*(a*b)^(1/2)*x^5*a*b^5+45045*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(13/2)*b^7-63525
*B*(a*b)^(1/2)*x^5*a^2*b^4+219219*A*(a*b)^(1/2)*x^4*a^2*b^4-84315*B*(a*b)^(1/2)*
x^4*a^3*b^3+119691*A*(a*b)^(1/2)*x^3*a^3*b^3+45045*A*arctan(x^(1/2)*b/(a*b)^(1/2
))*x^(5/2)*a^4*b^3-17325*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(13/2)*a*b^6+180180*A
*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(11/2)*a*b^6-46035*B*(a*b)^(1/2)*x^3*a^4*b^2-17
325*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(5/2)*a^5*b^2+18304*A*(a*b)^(1/2)*x^2*a^4*
b^2-7040*B*(a*b)^(1/2)*x^2*a^5*b-1664*A*(a*b)^(1/2)*x*a^5*b+45045*A*(a*b)^(1/2)*
x^6*b^6+640*B*(a*b)^(1/2)*x*a^6+384*A*(a*b)^(1/2)*a^6)*(b*x+a)/(a*b)^(1/2)/x^(5/
2)/a^7/((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)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294672, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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^(7/2)),x, algorithm="fricas")

[Out]

[-1/1920*(768*A*a^6 - 6930*(5*B*a*b^5 - 13*A*b^6)*x^6 - 25410*(5*B*a^2*b^4 - 13*
A*a*b^5)*x^5 - 33726*(5*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 - 18414*(5*B*a^4*b^2 - 13*
A*a^3*b^3)*x^3 - 2816*(5*B*a^5*b - 13*A*a^4*b^2)*x^2 + 3465*((5*B*a*b^5 - 13*A*b
^6)*x^6 + 4*(5*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 6*(5*B*a^3*b^3 - 13*A*a^2*b^4)*x^4
+ 4*(5*B*a^4*b^2 - 13*A*a^3*b^3)*x^3 + (5*B*a^5*b - 13*A*a^4*b^2)*x^2)*sqrt(x)*s
qrt(-b/a)*log((b*x - 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 256*(5*B*a^6 - 13*
A*a^5*b)*x)/((a^7*b^4*x^6 + 4*a^8*b^3*x^5 + 6*a^9*b^2*x^4 + 4*a^10*b*x^3 + a^11*
x^2)*sqrt(x)), -1/960*(384*A*a^6 - 3465*(5*B*a*b^5 - 13*A*b^6)*x^6 - 12705*(5*B*
a^2*b^4 - 13*A*a*b^5)*x^5 - 16863*(5*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 - 9207*(5*B*a
^4*b^2 - 13*A*a^3*b^3)*x^3 - 1408*(5*B*a^5*b - 13*A*a^4*b^2)*x^2 + 3465*((5*B*a*
b^5 - 13*A*b^6)*x^6 + 4*(5*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 6*(5*B*a^3*b^3 - 13*A*a
^2*b^4)*x^4 + 4*(5*B*a^4*b^2 - 13*A*a^3*b^3)*x^3 + (5*B*a^5*b - 13*A*a^4*b^2)*x^
2)*sqrt(x)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + 128*(5*B*a^6 - 13*A*a^5*b
)*x)/((a^7*b^4*x^6 + 4*a^8*b^3*x^5 + 6*a^9*b^2*x^4 + 4*a^10*b*x^3 + a^11*x^2)*sq
rt(x))]

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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**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27913, size = 279, normalized size = 0.69 \[ \frac{231 \,{\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} a^{7}{\rm sign}\left (b x + a\right )} + \frac{2 \,{\left (75 \, B a b x^{2} - 225 \, A b^{2} x^{2} - 5 \, B a^{2} x + 25 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{7} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right )} + \frac{1545 \, B a b^{5} x^{\frac{7}{2}} - 3249 \, A b^{6} x^{\frac{7}{2}} + 5153 \, B a^{2} b^{4} x^{\frac{5}{2}} - 10633 \, A a b^{5} x^{\frac{5}{2}} + 5855 \, B a^{3} b^{3} x^{\frac{3}{2}} - 11767 \, A a^{2} b^{4} x^{\frac{3}{2}} + 2295 \, B a^{4} b^{2} \sqrt{x} - 4431 \, A a^{3} b^{3} \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} a^{7}{\rm sign}\left (b x + a\right )} \]

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)*x^(7/2)),x, algorithm="giac")

[Out]

231/64*(5*B*a*b^2 - 13*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7*sign(b*
x + a)) + 2/15*(75*B*a*b*x^2 - 225*A*b^2*x^2 - 5*B*a^2*x + 25*A*a*b*x - 3*A*a^2)
/(a^7*x^(5/2)*sign(b*x + a)) + 1/192*(1545*B*a*b^5*x^(7/2) - 3249*A*b^6*x^(7/2)
+ 5153*B*a^2*b^4*x^(5/2) - 10633*A*a*b^5*x^(5/2) + 5855*B*a^3*b^3*x^(3/2) - 1176
7*A*a^2*b^4*x^(3/2) + 2295*B*a^4*b^2*sqrt(x) - 4431*A*a^3*b^3*sqrt(x))/((b*x + a
)^4*a^7*sign(b*x + a))