3.2137 \(\int \frac{a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=264 \[ -\frac{7 b^2 e (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}-\frac{7 b e (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}-\frac{7 e (a+b x)}{5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac{7 b^{5/2} e (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \]

[Out]

-(1/((b*d - a*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (7*e*(a + b*x
))/(5*(b*d - a*e)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*b*e*(a +
b*x))/(3*(b*d - a*e)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*b^2*
e*(a + b*x))/((b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*b^
(5/2)*e*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/((b*d - a*e)
^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.416927, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{7 b^2 e (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}-\frac{7 b e (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}-\frac{7 e (a+b x)}{5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac{7 b^{5/2} e (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

-(1/((b*d - a*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (7*e*(a + b*x
))/(5*(b*d - a*e)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*b*e*(a +
b*x))/(3*(b*d - a*e)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*b^2*
e*(a + b*x))/((b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*b^
(5/2)*e*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/((b*d - a*e)
^(9/2)*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**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.983107, size = 154, normalized size = 0.58 \[ \frac{(a+b x) \left (\frac{7 b^{5/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{9/2}}-\frac{\sqrt{d+e x} \left (\frac{15 b^3}{a+b x}+\frac{20 b e (b d-a e)}{(d+e x)^2}+\frac{6 e (b d-a e)^2}{(d+e x)^3}+\frac{90 b^2 e}{d+e x}\right )}{15 (b d-a e)^4}\right )}{\sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

((a + b*x)*(-(Sqrt[d + e*x]*((15*b^3)/(a + b*x) + (6*e*(b*d - a*e)^2)/(d + e*x)^
3 + (20*b*e*(b*d - a*e))/(d + e*x)^2 + (90*b^2*e)/(d + e*x)))/(15*(b*d - a*e)^4)
+ (7*b^(5/2)*e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(9
/2)))/Sqrt[(a + b*x)^2]

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Maple [A]  time = 0.03, size = 343, normalized size = 1.3 \[ -{\frac{ \left ( bx+a \right ) ^{2}}{15\, \left ( ae-bd \right ) ^{4}} \left ( 105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{5/2}x{b}^{4}e+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{5/2}a{b}^{3}e+105\,\sqrt{b \left ( ae-bd \right ) }{x}^{3}{b}^{3}{e}^{3}+70\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}a{b}^{2}{e}^{3}+245\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{3}d{e}^{2}-14\,\sqrt{b \left ( ae-bd \right ) }x{a}^{2}b{e}^{3}+168\,\sqrt{b \left ( ae-bd \right ) }xa{b}^{2}d{e}^{2}+161\,\sqrt{b \left ( ae-bd \right ) }x{b}^{3}{d}^{2}e+6\,\sqrt{b \left ( ae-bd \right ) }{a}^{3}{e}^{3}-32\,\sqrt{b \left ( ae-bd \right ) }{a}^{2}bd{e}^{2}+116\,\sqrt{b \left ( ae-bd \right ) }a{b}^{2}{d}^{2}e+15\,\sqrt{b \left ( ae-bd \right ) }{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( ex+d \right ) ^{-{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

-1/15*(105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*x*b^4*e+105
*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*a*b^3*e+105*(b*(a*e-b
*d))^(1/2)*x^3*b^3*e^3+70*(b*(a*e-b*d))^(1/2)*x^2*a*b^2*e^3+245*(b*(a*e-b*d))^(1
/2)*x^2*b^3*d*e^2-14*(b*(a*e-b*d))^(1/2)*x*a^2*b*e^3+168*(b*(a*e-b*d))^(1/2)*x*a
*b^2*d*e^2+161*(b*(a*e-b*d))^(1/2)*x*b^3*d^2*e+6*(b*(a*e-b*d))^(1/2)*a^3*e^3-32*
(b*(a*e-b*d))^(1/2)*a^2*b*d*e^2+116*(b*(a*e-b*d))^(1/2)*a*b^2*d^2*e+15*(b*(a*e-b
*d))^(1/2)*b^3*d^3)*(b*x+a)^2/(b*(a*e-b*d))^(1/2)/(e*x+d)^(5/2)/(a*e-b*d)^4/((b*
x+a)^2)^(3/2)

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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)^(3/2)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.319805, 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)^(3/2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[-1/30*(210*b^3*e^3*x^3 + 30*b^3*d^3 + 232*a*b^2*d^2*e - 64*a^2*b*d*e^2 + 12*a^3
*e^3 + 70*(7*b^3*d*e^2 + 2*a*b^2*e^3)*x^2 - 105*(b^3*e^3*x^3 + a*b^2*d^2*e + (2*
b^3*d*e^2 + a*b^2*e^3)*x^2 + (b^3*d^2*e + 2*a*b^2*d*e^2)*x)*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)) + 14*(23*b^3*d^2*e + 24*a*b^2*d*e^2 - 2*a^2*b*e^3)*x)/((a*b
^4*d^6 - 4*a^2*b^3*d^5*e + 6*a^3*b^2*d^4*e^2 - 4*a^4*b*d^3*e^3 + a^5*d^2*e^4 + (
b^5*d^4*e^2 - 4*a*b^4*d^3*e^3 + 6*a^2*b^3*d^2*e^4 - 4*a^3*b^2*d*e^5 + a^4*b*e^6)
*x^3 + (2*b^5*d^5*e - 7*a*b^4*d^4*e^2 + 8*a^2*b^3*d^3*e^3 - 2*a^3*b^2*d^2*e^4 -
2*a^4*b*d*e^5 + a^5*e^6)*x^2 + (b^5*d^6 - 2*a*b^4*d^5*e - 2*a^2*b^3*d^4*e^2 + 8*
a^3*b^2*d^3*e^3 - 7*a^4*b*d^2*e^4 + 2*a^5*d*e^5)*x)*sqrt(e*x + d)), -1/15*(105*b
^3*e^3*x^3 + 15*b^3*d^3 + 116*a*b^2*d^2*e - 32*a^2*b*d*e^2 + 6*a^3*e^3 + 35*(7*b
^3*d*e^2 + 2*a*b^2*e^3)*x^2 - 105*(b^3*e^3*x^3 + a*b^2*d^2*e + (2*b^3*d*e^2 + a*
b^2*e^3)*x^2 + (b^3*d^2*e + 2*a*b^2*d*e^2)*x)*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)) + 7*(23*b^3*d^2*e +
24*a*b^2*d*e^2 - 2*a^2*b*e^3)*x)/((a*b^4*d^6 - 4*a^2*b^3*d^5*e + 6*a^3*b^2*d^4*
e^2 - 4*a^4*b*d^3*e^3 + a^5*d^2*e^4 + (b^5*d^4*e^2 - 4*a*b^4*d^3*e^3 + 6*a^2*b^3
*d^2*e^4 - 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^3 + (2*b^5*d^5*e - 7*a*b^4*d^4*e^2 + 8
*a^2*b^3*d^3*e^3 - 2*a^3*b^2*d^2*e^4 - 2*a^4*b*d*e^5 + a^5*e^6)*x^2 + (b^5*d^6 -
2*a*b^4*d^5*e - 2*a^2*b^3*d^4*e^2 + 8*a^3*b^2*d^3*e^3 - 7*a^4*b*d^2*e^4 + 2*a^5
*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**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.32664, size = 883, normalized size = 3.34 \[ \frac{7 \, b^{3} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{{\left (b^{4} d^{4} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{\sqrt{x e + d} b^{3} e^{2}}{{\left (b^{4} d^{4} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}} + \frac{2 \,{\left (45 \,{\left (x e + d\right )}^{2} b^{2} e^{2} + 10 \,{\left (x e + d\right )} b^{2} d e^{2} + 3 \, b^{2} d^{2} e^{2} - 10 \,{\left (x e + d\right )} a b e^{3} - 6 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )}}{15 \,{\left (b^{4} d^{4} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left (x e + d\right )}^{\frac{5}{2}}} \]

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)^(3/2)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

7*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^2/((b^4*d^4*e*sign(-(x*e +
d)*b*e + b*d*e - a*e^2) - 4*a*b^3*d^3*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) +
6*a^2*b^2*d^2*e^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^4*sign(-(x
*e + d)*b*e + b*d*e - a*e^2) + a^4*e^5*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sqr
t(-b^2*d + a*b*e)) + sqrt(x*e + d)*b^3*e^2/((b^4*d^4*e*sign(-(x*e + d)*b*e + b*d
*e - a*e^2) - 4*a*b^3*d^3*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 6*a^2*b^2*d
^2*e^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^4*sign(-(x*e + d)*b*e
+ b*d*e - a*e^2) + a^4*e^5*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*((x*e + d)*b -
b*d + a*e)) + 2/15*(45*(x*e + d)^2*b^2*e^2 + 10*(x*e + d)*b^2*d*e^2 + 3*b^2*d^2*
e^2 - 10*(x*e + d)*a*b*e^3 - 6*a*b*d*e^3 + 3*a^2*e^4)/((b^4*d^4*e*sign(-(x*e + d
)*b*e + b*d*e - a*e^2) - 4*a*b^3*d^3*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) +
6*a^2*b^2*d^2*e^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^4*sign(-(x*
e + d)*b*e + b*d*e - a*e^2) + a^4*e^5*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*(x*e
+ d)^(5/2))