3.1865 \(\int \frac{A+B x}{(d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=251 \[ \frac{2 b (a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{2 (a+b x) (A b-a B)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{2 (a+b x) (B d-A e)}{5 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{2 b^{3/2} (a+b x) (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

[Out]

(-2*(B*d - A*e)*(a + b*x))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (2*(A*b - a*B)*(a + b*x))/(3*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) + (2*b*(A*b - a*B)*(a + b*x))/((b*d - a*e)^3*Sqrt[d + e*x
]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*b^(3/2)*(A*b - a*B)*(a + b*x)*ArcTanh[(Sqr
t[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/((b*d - a*e)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2])

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Rubi [A]  time = 0.433071, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 b (a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{2 (a+b x) (A b-a B)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{2 (a+b x) (B d-A e)}{5 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{2 b^{3/2} (a+b x) (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(B*d - A*e)*(a + b*x))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (2*(A*b - a*B)*(a + b*x))/(3*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) + (2*b*(A*b - a*B)*(a + b*x))/((b*d - a*e)^3*Sqrt[d + e*x
]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*b^(3/2)*(A*b - a*B)*(a + b*x)*ArcTanh[(Sqr
t[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/((b*d - a*e)^(7/2)*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)/(e*x+d)**(7/2)/((b*x+a)**2)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.832802, size = 168, normalized size = 0.67 \[ \frac{(a+b x) \left (-\frac{2 b^{3/2} (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}+\frac{2 b (A b-a B)}{\sqrt{d+e x} (b d-a e)^3}+\frac{2 (A b-a B)}{3 (d+e x)^{3/2} (b d-a e)^2}-\frac{2 (A e-B d)}{5 e (d+e x)^{5/2} (a e-b d)}\right )}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*((-2*(-(B*d) + A*e))/(5*e*(-(b*d) + a*e)*(d + e*x)^(5/2)) + (2*(A*b -
a*B))/(3*(b*d - a*e)^2*(d + e*x)^(3/2)) + (2*b*(A*b - a*B))/((b*d - a*e)^3*Sqrt
[d + e*x]) - (2*b^(3/2)*(A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a
*e]])/(b*d - a*e)^(7/2)))/Sqrt[(a + b*x)^2]

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Maple [B]  time = 0.026, size = 386, normalized size = 1.5 \[ -{\frac{2\,bx+2\,a}{15\,e \left ( ae-bd \right ) ^{3}} \left ( 15\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{5/2}{b}^{3}e-15\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{5/2}a{b}^{2}e+15\,A\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{2}{e}^{3}-15\,B\sqrt{b \left ( ae-bd \right ) }{x}^{2}ab{e}^{3}-5\,A\sqrt{b \left ( ae-bd \right ) }xab{e}^{3}+35\,A\sqrt{b \left ( ae-bd \right ) }x{b}^{2}d{e}^{2}+5\,B\sqrt{b \left ( ae-bd \right ) }x{a}^{2}{e}^{3}-35\,B\sqrt{b \left ( ae-bd \right ) }xabd{e}^{2}+3\,A\sqrt{b \left ( ae-bd \right ) }{a}^{2}{e}^{3}-11\,A\sqrt{b \left ( ae-bd \right ) }abd{e}^{2}+23\,A\sqrt{b \left ( ae-bd \right ) }{b}^{2}{d}^{2}e+2\,B\sqrt{b \left ( ae-bd \right ) }{a}^{2}d{e}^{2}-14\,B\sqrt{b \left ( ae-bd \right ) }ab{d}^{2}e-3\,B\sqrt{b \left ( ae-bd \right ) }{b}^{2}{d}^{3} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(7/2)/((b*x+a)^2)^(1/2),x)

[Out]

-2/15*(b*x+a)*(15*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*b^
3*e-15*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*a*b^2*e+15*A*
(b*(a*e-b*d))^(1/2)*x^2*b^2*e^3-15*B*(b*(a*e-b*d))^(1/2)*x^2*a*b*e^3-5*A*(b*(a*e
-b*d))^(1/2)*x*a*b*e^3+35*A*(b*(a*e-b*d))^(1/2)*x*b^2*d*e^2+5*B*(b*(a*e-b*d))^(1
/2)*x*a^2*e^3-35*B*(b*(a*e-b*d))^(1/2)*x*a*b*d*e^2+3*A*(b*(a*e-b*d))^(1/2)*a^2*e
^3-11*A*(b*(a*e-b*d))^(1/2)*a*b*d*e^2+23*A*(b*(a*e-b*d))^(1/2)*b^2*d^2*e+2*B*(b*
(a*e-b*d))^(1/2)*a^2*d*e^2-14*B*(b*(a*e-b*d))^(1/2)*a*b*d^2*e-3*B*(b*(a*e-b*d))^
(1/2)*b^2*d^3)/((b*x+a)^2)^(1/2)/e/(a*e-b*d)^3/(e*x+d)^(5/2)/(b*(a*e-b*d))^(1/2)

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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)/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295729, size = 1, normalized size = 0. \[ \left [-\frac{6 \, B b^{2} d^{3} - 6 \, A a^{2} e^{3} + 30 \,{\left (B a b - A b^{2}\right )} e^{3} x^{2} + 2 \,{\left (14 \, B a b - 23 \, A b^{2}\right )} d^{2} e - 2 \,{\left (2 \, B a^{2} - 11 \, A a b\right )} d e^{2} - 15 \,{\left ({\left (B a b - A b^{2}\right )} e^{3} x^{2} + 2 \,{\left (B a b - A b^{2}\right )} d e^{2} x +{\left (B a b - A b^{2}\right )} d^{2} e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 10 \,{\left (7 \,{\left (B a b - A b^{2}\right )} d e^{2} -{\left (B a^{2} - A a b\right )} e^{3}\right )} x}{15 \,{\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4} +{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{2} + 2 \,{\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x\right )} \sqrt{e x + d}}, -\frac{2 \,{\left (3 \, B b^{2} d^{3} - 3 \, A a^{2} e^{3} + 15 \,{\left (B a b - A b^{2}\right )} e^{3} x^{2} +{\left (14 \, B a b - 23 \, A b^{2}\right )} d^{2} e -{\left (2 \, B a^{2} - 11 \, A a b\right )} d e^{2} - 15 \,{\left ({\left (B a b - A b^{2}\right )} e^{3} x^{2} + 2 \,{\left (B a b - A b^{2}\right )} d e^{2} x +{\left (B a b - A b^{2}\right )} d^{2} e\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) + 5 \,{\left (7 \,{\left (B a b - A b^{2}\right )} d e^{2} -{\left (B a^{2} - A a b\right )} e^{3}\right )} x\right )}}{15 \,{\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4} +{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{2} + 2 \,{\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[-1/15*(6*B*b^2*d^3 - 6*A*a^2*e^3 + 30*(B*a*b - A*b^2)*e^3*x^2 + 2*(14*B*a*b - 2
3*A*b^2)*d^2*e - 2*(2*B*a^2 - 11*A*a*b)*d*e^2 - 15*((B*a*b - A*b^2)*e^3*x^2 + 2*
(B*a*b - A*b^2)*d*e^2*x + (B*a*b - A*b^2)*d^2*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e
))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(
b*x + a)) + 10*(7*(B*a*b - A*b^2)*d*e^2 - (B*a^2 - A*a*b)*e^3)*x)/((b^3*d^5*e -
3*a*b^2*d^4*e^2 + 3*a^2*b*d^3*e^3 - a^3*d^2*e^4 + (b^3*d^3*e^3 - 3*a*b^2*d^2*e^4
+ 3*a^2*b*d*e^5 - a^3*e^6)*x^2 + 2*(b^3*d^4*e^2 - 3*a*b^2*d^3*e^3 + 3*a^2*b*d^2
*e^4 - a^3*d*e^5)*x)*sqrt(e*x + d)), -2/15*(3*B*b^2*d^3 - 3*A*a^2*e^3 + 15*(B*a*
b - A*b^2)*e^3*x^2 + (14*B*a*b - 23*A*b^2)*d^2*e - (2*B*a^2 - 11*A*a*b)*d*e^2 -
15*((B*a*b - A*b^2)*e^3*x^2 + 2*(B*a*b - A*b^2)*d*e^2*x + (B*a*b - A*b^2)*d^2*e)
*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sq
rt(e*x + d)*b)) + 5*(7*(B*a*b - A*b^2)*d*e^2 - (B*a^2 - A*a*b)*e^3)*x)/((b^3*d^5
*e - 3*a*b^2*d^4*e^2 + 3*a^2*b*d^3*e^3 - a^3*d^2*e^4 + (b^3*d^3*e^3 - 3*a*b^2*d^
2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6)*x^2 + 2*(b^3*d^4*e^2 - 3*a*b^2*d^3*e^3 + 3*a^2*
b*d^2*e^4 - a^3*d*e^5)*x)*sqrt(e*x + d))]

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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)/(e*x+d)**(7/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.309961, size = 497, normalized size = 1.98 \[ -\frac{2 \,{\left (B a b^{2}{\rm sign}\left (b x + a\right ) - A b^{3}{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (3 \, B b^{2} d^{3}{\rm sign}\left (b x + a\right ) + 15 \,{\left (x e + d\right )}^{2} B a b e{\rm sign}\left (b x + a\right ) - 15 \,{\left (x e + d\right )}^{2} A b^{2} e{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )} B a b d e{\rm sign}\left (b x + a\right ) - 5 \,{\left (x e + d\right )} A b^{2} d e{\rm sign}\left (b x + a\right ) - 6 \, B a b d^{2} e{\rm sign}\left (b x + a\right ) - 3 \, A b^{2} d^{2} e{\rm sign}\left (b x + a\right ) - 5 \,{\left (x e + d\right )} B a^{2} e^{2}{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )} A a b e^{2}{\rm sign}\left (b x + a\right ) + 3 \, B a^{2} d e^{2}{\rm sign}\left (b x + a\right ) + 6 \, A a b d e^{2}{\rm sign}\left (b x + a\right ) - 3 \, A a^{2} e^{3}{\rm sign}\left (b x + a\right )\right )}}{15 \,{\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

-2*(B*a*b^2*sign(b*x + a) - A*b^3*sign(b*x + a))*arctan(sqrt(x*e + d)*b/sqrt(-b^
2*d + a*b*e))/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-b^2*d +
a*b*e)) - 2/15*(3*B*b^2*d^3*sign(b*x + a) + 15*(x*e + d)^2*B*a*b*e*sign(b*x + a
) - 15*(x*e + d)^2*A*b^2*e*sign(b*x + a) + 5*(x*e + d)*B*a*b*d*e*sign(b*x + a) -
5*(x*e + d)*A*b^2*d*e*sign(b*x + a) - 6*B*a*b*d^2*e*sign(b*x + a) - 3*A*b^2*d^2
*e*sign(b*x + a) - 5*(x*e + d)*B*a^2*e^2*sign(b*x + a) + 5*(x*e + d)*A*a*b*e^2*s
ign(b*x + a) + 3*B*a^2*d*e^2*sign(b*x + a) + 6*A*a*b*d*e^2*sign(b*x + a) - 3*A*a
^2*e^3*sign(b*x + a))/((b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^4)*(
x*e + d)^(5/2))