3.1864 \(\int \frac{A+B x}{(d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=194 \[ \frac{2 (a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{2 (a+b x) (B d-A e)}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{2 \sqrt{b} (a+b x) (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.377569, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 (a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{2 (a+b x) (B d-A e)}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{2 \sqrt{b} (a+b x) (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.577602, size = 135, normalized size = 0.7 \[ \frac{(a+b x) \left (-\frac{2 \left (a e (A e+2 B d+3 B e x)-A b e (4 d+3 e x)+b B d^2\right )}{3 e (d+e x)^{3/2} (b d-a e)^2}-\frac{2 \sqrt{b} (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}}\right )}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*((-2*(b*B*d^2 - A*b*e*(4*d + 3*e*x) + a*e*(2*B*d + A*e + 3*B*e*x)))/(
3*e*(b*d - a*e)^2*(d + e*x)^(3/2)) - (2*Sqrt[b]*(A*b - a*B)*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(5/2)))/Sqrt[(a + b*x)^2]

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Maple [A]  time = 0.023, size = 235, normalized size = 1.2 \[{\frac{2\,bx+2\,a}{3\,e \left ( ae-bd \right ) ^{2}} \left ( 3\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}{b}^{2}e-3\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}abe+3\,A\sqrt{b \left ( ae-bd \right ) }xb{e}^{2}-3\,B\sqrt{b \left ( ae-bd \right ) }xa{e}^{2}-A\sqrt{b \left ( ae-bd \right ) }a{e}^{2}+4\,A\sqrt{b \left ( ae-bd \right ) }bde-2\,B\sqrt{b \left ( ae-bd \right ) }ade-B\sqrt{b \left ( ae-bd \right ) }b{d}^{2} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2),x)

[Out]

2/3*(b*x+a)*(3*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*b^2*e
-3*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*a*b*e+3*A*(b*(a*e
-b*d))^(1/2)*x*b*e^2-3*B*(b*(a*e-b*d))^(1/2)*x*a*e^2-A*(b*(a*e-b*d))^(1/2)*a*e^2
+4*A*(b*(a*e-b*d))^(1/2)*b*d*e-2*B*(b*(a*e-b*d))^(1/2)*a*d*e-B*(b*(a*e-b*d))^(1/
2)*b*d^2)/((b*x+a)^2)^(1/2)/e/(a*e-b*d)^2/(e*x+d)^(3/2)/(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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294603, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, B b d^{2} + 2 \, A a e^{2} + 6 \,{\left (B a - A b\right )} e^{2} x + 4 \,{\left (B a - 2 \, A b\right )} d e + 3 \,{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right )}{3 \,{\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}, -\frac{2 \,{\left (B b d^{2} + A a e^{2} + 3 \,{\left (B a - A b\right )} e^{2} x + 2 \,{\left (B a - 2 \, A b\right )} d e - 3 \,{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right )\right )}}{3 \,{\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[-1/3*(2*B*b*d^2 + 2*A*a*e^2 + 6*(B*a - A*b)*e^2*x + 4*(B*a - 2*A*b)*d*e + 3*((B
*a - A*b)*e^2*x + (B*a - A*b)*d*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)))/((b
^2*d^3*e - 2*a*b*d^2*e^2 + a^2*d*e^3 + (b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*x)*
sqrt(e*x + d)), -2/3*(B*b*d^2 + A*a*e^2 + 3*(B*a - A*b)*e^2*x + 2*(B*a - 2*A*b)*
d*e - 3*((B*a - A*b)*e^2*x + (B*a - A*b)*d*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))
*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)*b)))/((b^2*d^3*e - 2*a*
b*d^2*e^2 + a^2*d*e^3 + (b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*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)**(5/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.301267, size = 282, normalized size = 1.45 \[ -\frac{2 \,{\left (B a b{\rm sign}\left (b x + a\right ) - A b^{2}{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (B b d^{2}{\rm sign}\left (b x + a\right ) + 3 \,{\left (x e + d\right )} B a e{\rm sign}\left (b x + a\right ) - 3 \,{\left (x e + d\right )} A b e{\rm sign}\left (b x + a\right ) - B a d e{\rm sign}\left (b x + a\right ) - A b d e{\rm sign}\left (b x + a\right ) + A a e^{2}{\rm sign}\left (b x + a\right )\right )}}{3 \,{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

-2*(B*a*b*sign(b*x + a) - A*b^2*sign(b*x + a))*arctan(sqrt(x*e + d)*b/sqrt(-b^2*
d + a*b*e))/((b^2*d^2 - 2*a*b*d*e + a^2*e^2)*sqrt(-b^2*d + a*b*e)) - 2/3*(B*b*d^
2*sign(b*x + a) + 3*(x*e + d)*B*a*e*sign(b*x + a) - 3*(x*e + d)*A*b*e*sign(b*x +
a) - B*a*d*e*sign(b*x + a) - A*b*d*e*sign(b*x + a) + A*a*e^2*sign(b*x + a))/((b
^2*d^2*e - 2*a*b*d*e^2 + a^2*e^3)*(x*e + d)^(3/2))