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

Optimal. Leaf size=306 \[ \frac{x^{9/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{7/2} (A b-9 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 (a+b x) (A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 \sqrt{a} b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 \sqrt{x} (a+b x) (A b-9 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 x^{3/2} (A b-9 a B)}{192 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 x^{5/2} (A b-9 a B)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(35*(A*b - 9*a*B)*x^(3/2))/(192*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a
*B)*x^(9/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - 9*a*B)*
x^(7/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*(A*b - 9*a*B)
*x^(5/2))/(96*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(A*b - 9*a*B)
*Sqrt[x]*(a + b*x))/(64*a*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*(A*b - 9*a*B)
*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*Sqrt[a]*b^(11/2)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])

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Rubi [A]  time = 0.406791, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{x^{9/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{7/2} (A b-9 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 (a+b x) (A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 \sqrt{a} b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 \sqrt{x} (a+b x) (A b-9 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 x^{3/2} (A b-9 a B)}{192 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 x^{5/2} (A b-9 a B)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(35*(A*b - 9*a*B)*x^(3/2))/(192*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a
*B)*x^(9/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - 9*a*B)*
x^(7/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*(A*b - 9*a*B)
*x^(5/2))/(96*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(A*b - 9*a*B)
*Sqrt[x]*(a + b*x))/(64*a*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*(A*b - 9*a*B)
*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*Sqrt[a]*b^(11/2)*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(x**(7/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.199624, size = 151, normalized size = 0.49 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{x} \left (945 a^4 B-105 a^3 b (A-33 B x)+7 a^2 b^2 x (657 B x-55 A)+a b^3 x^2 (2511 B x-511 A)+3 b^4 x^3 (128 B x-93 A)\right )+105 (a+b x)^4 (A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{192 \sqrt{a} b^{11/2} (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*Sqrt[b]*Sqrt[x]*(945*a^4*B - 105*a^3*b*(A - 33*B*x) + 3*b^4*x^3*(-93*A
+ 128*B*x) + 7*a^2*b^2*x*(-55*A + 657*B*x) + a*b^3*x^2*(-511*A + 2511*B*x)) + 10
5*(A*b - 9*a*B)*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(192*Sqrt[a]*b^(1
1/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.029, size = 368, normalized size = 1.2 \[ -{\frac{bx+a}{192\,{b}^{5}} \left ( 279\,A\sqrt{ab}{x}^{7/2}{b}^{4}-2511\,B\sqrt{ab}{x}^{7/2}a{b}^{3}+511\,A\sqrt{ab}{x}^{5/2}a{b}^{3}-105\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}{b}^{5}-4599\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}-384\,B\sqrt{ab}{x}^{9/2}{b}^{4}+945\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}a{b}^{4}-420\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}a{b}^{4}+3780\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{3}+385\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}-630\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{3}-3465\,B\sqrt{ab}{x}^{3/2}{a}^{3}b+5670\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{2}-420\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{3}{b}^{2}+3780\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{4}b+105\,A\sqrt{ab}\sqrt{x}{a}^{3}b-105\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4}b-945\,B\sqrt{ab}\sqrt{x}{a}^{4}+945\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*(279*A*(a*b)^(1/2)*x^(7/2)*b^4-2511*B*(a*b)^(1/2)*x^(7/2)*a*b^3+511*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^5-4599*B*(a*b)^
(1/2)*x^(5/2)*a^2*b^2-384*B*(a*b)^(1/2)*x^(9/2)*b^4+945*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+3780*B*arctan(x^
(1/2)*b/(a*b)^(1/2))*x^3*a^2*b^3+385*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^3-3465*B*(a*b)^(1/2)*x^(3/2)*a^3*b+5670*B*arcta
n(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+3780*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^4*b+105*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-945*B*(a*b)^(1/2)*x^(1/2)*a^4+945*B*ar
ctan(x^(1/2)*b/(a*b)^(1/2))*a^5)*(b*x+a)/(a*b)^(1/2)/b^5/((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)*x^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29746, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (384 \, B b^{4} x^{4} + 945 \, B a^{4} - 105 \, A a^{3} b + 279 \,{\left (9 \, B a b^{3} - A b^{4}\right )} x^{3} + 511 \,{\left (9 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 385 \,{\left (9 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 105 \,{\left (9 \, B a^{5} - A a^{4} b +{\left (9 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \,{\left (9 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \,{\left (9 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (9 \, B a^{4} b - A a^{3} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{384 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )} \sqrt{-a b}}, \frac{{\left (384 \, B b^{4} x^{4} + 945 \, B a^{4} - 105 \, A a^{3} b + 279 \,{\left (9 \, B a b^{3} - A b^{4}\right )} x^{3} + 511 \,{\left (9 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 385 \,{\left (9 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 105 \,{\left (9 \, B a^{5} - A a^{4} b +{\left (9 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \,{\left (9 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \,{\left (9 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (9 \, B a^{4} b - A a^{3} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{192 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )} \sqrt{a b}}\right ] \]

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, algorithm="fricas")

[Out]

[1/384*(2*(384*B*b^4*x^4 + 945*B*a^4 - 105*A*a^3*b + 279*(9*B*a*b^3 - A*b^4)*x^3
+ 511*(9*B*a^2*b^2 - A*a*b^3)*x^2 + 385*(9*B*a^3*b - A*a^2*b^2)*x)*sqrt(-a*b)*s
qrt(x) - 105*(9*B*a^5 - A*a^4*b + (9*B*a*b^4 - A*b^5)*x^4 + 4*(9*B*a^2*b^3 - A*a
*b^4)*x^3 + 6*(9*B*a^3*b^2 - A*a^2*b^3)*x^2 + 4*(9*B*a^4*b - A*a^3*b^2)*x)*log((
2*a*b*sqrt(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((b^9*x^4 + 4*a*b^8*x^3 + 6*a^
2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5)*sqrt(-a*b)), 1/192*((384*B*b^4*x^4 + 945*B*a^
4 - 105*A*a^3*b + 279*(9*B*a*b^3 - A*b^4)*x^3 + 511*(9*B*a^2*b^2 - A*a*b^3)*x^2
+ 385*(9*B*a^3*b - A*a^2*b^2)*x)*sqrt(a*b)*sqrt(x) + 105*(9*B*a^5 - A*a^4*b + (9
*B*a*b^4 - A*b^5)*x^4 + 4*(9*B*a^2*b^3 - A*a*b^4)*x^3 + 6*(9*B*a^3*b^2 - A*a^2*b
^3)*x^2 + 4*(9*B*a^4*b - A*a^3*b^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x))))/((b^9*x^4
+ 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5)*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(x**(7/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278897, size = 215, normalized size = 0.7 \[ \frac{2 \, B \sqrt{x}}{b^{5}{\rm sign}\left (b x + a\right )} - \frac{35 \,{\left (9 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} b^{5}{\rm sign}\left (b x + a\right )} + \frac{975 \, B a b^{3} x^{\frac{7}{2}} - 279 \, A b^{4} x^{\frac{7}{2}} + 2295 \, B a^{2} b^{2} x^{\frac{5}{2}} - 511 \, A a b^{3} x^{\frac{5}{2}} + 1929 \, B a^{3} b x^{\frac{3}{2}} - 385 \, A a^{2} b^{2} x^{\frac{3}{2}} + 561 \, B a^{4} \sqrt{x} - 105 \, A a^{3} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} b^{5}{\rm sign}\left (b x + a\right )} \]

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, algorithm="giac")

[Out]

2*B*sqrt(x)/(b^5*sign(b*x + a)) - 35/64*(9*B*a - A*b)*arctan(b*sqrt(x)/sqrt(a*b)
)/(sqrt(a*b)*b^5*sign(b*x + a)) + 1/192*(975*B*a*b^3*x^(7/2) - 279*A*b^4*x^(7/2)
+ 2295*B*a^2*b^2*x^(5/2) - 511*A*a*b^3*x^(5/2) + 1929*B*a^3*b*x^(3/2) - 385*A*a
^2*b^2*x^(3/2) + 561*B*a^4*sqrt(x) - 105*A*a^3*b*sqrt(x))/((b*x + a)^4*b^5*sign(
b*x + a))