3.2143 \(\int \frac{a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=276 \[ -\frac{35 e^3 (a+b x)}{8 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}+\frac{35 \sqrt{b} e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}-\frac{35 e^2}{24 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{7 e}{12 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

[Out]

(-35*e^2)/(24*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(3*
(b*d - a*e)*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e)/(12
*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*
(a + b*x))/(8*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*S
qrt[b]*e^3*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d -
a*e)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.480041, antiderivative size = 276, 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 \[ -\frac{35 e^3 (a+b x)}{8 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}+\frac{35 \sqrt{b} e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}-\frac{35 e^2}{24 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{7 e}{12 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-35*e^2)/(24*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(3*
(b*d - a*e)*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e)/(12
*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*
(a + b*x))/(8*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*S
qrt[b]*e^3*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.821273, size = 159, normalized size = 0.58 \[ \frac{(a+b x) \left (\frac{35 \sqrt{b} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{9/2}}-\frac{\sqrt{d+e x} \left (-\frac{22 b e (b d-a e)}{(a+b x)^2}+\frac{8 b (b d-a e)^2}{(a+b x)^3}+\frac{57 b e^2}{a+b x}+\frac{48 e^3}{d+e x}\right )}{3 (b d-a e)^4}\right )}{8 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

((a + b*x)*(-(Sqrt[d + e*x]*((8*b*(b*d - a*e)^2)/(a + b*x)^3 - (22*b*e*(b*d - a*
e))/(a + b*x)^2 + (57*b*e^2)/(a + b*x) + (48*e^3)/(d + e*x)))/(3*(b*d - a*e)^4)
+ (35*Sqrt[b]*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^
(9/2)))/(8*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.034, size = 431, normalized size = 1.6 \[ -{\frac{ \left ( bx+a \right ) ^{2}}{24\, \left ( ae-bd \right ) ^{4}} \left ( 105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{3}{b}^{4}{e}^{3}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{2}a{b}^{3}{e}^{3}+105\,\sqrt{b \left ( ae-bd \right ) }{x}^{3}{b}^{3}{e}^{3}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}x{a}^{2}{b}^{2}{e}^{3}+280\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}a{b}^{2}{e}^{3}+35\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{3}d{e}^{2}+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{a}^{3}b{e}^{3}+231\,\sqrt{b \left ( ae-bd \right ) }x{a}^{2}b{e}^{3}+98\,\sqrt{b \left ( ae-bd \right ) }xa{b}^{2}d{e}^{2}-14\,\sqrt{b \left ( ae-bd \right ) }x{b}^{3}{d}^{2}e+48\,\sqrt{b \left ( ae-bd \right ) }{a}^{3}{e}^{3}+87\,\sqrt{b \left ( ae-bd \right ) }{a}^{2}bd{e}^{2}-38\,\sqrt{b \left ( ae-bd \right ) }a{b}^{2}{d}^{2}e+8\,\sqrt{b \left ( ae-bd \right ) }{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/24*(105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x^3*b^4*e^3
+315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x^2*a*b^3*e^3+105
*(b*(a*e-b*d))^(1/2)*x^3*b^3*e^3+315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*(e*x+d)^(1/2)*x*a^2*b^2*e^3+280*(b*(a*e-b*d))^(1/2)*x^2*a*b^2*e^3+35*(b*(a*e-b*
d))^(1/2)*x^2*b^3*d*e^2+105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^
(1/2)*a^3*b*e^3+231*(b*(a*e-b*d))^(1/2)*x*a^2*b*e^3+98*(b*(a*e-b*d))^(1/2)*x*a*b
^2*d*e^2-14*(b*(a*e-b*d))^(1/2)*x*b^3*d^2*e+48*(b*(a*e-b*d))^(1/2)*a^3*e^3+87*(b
*(a*e-b*d))^(1/2)*a^2*b*d*e^2-38*(b*(a*e-b*d))^(1/2)*a*b^2*d^2*e+8*(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)^(1/2)/(a*e-b*d)^4/((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)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.337132, 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)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[-1/48*(210*b^3*e^3*x^3 + 16*b^3*d^3 - 76*a*b^2*d^2*e + 174*a^2*b*d*e^2 + 96*a^3
*e^3 + 70*(b^3*d*e^2 + 8*a*b^2*e^3)*x^2 - 105*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3
*a^2*b*e^3*x + a^3*e^3)*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*(2*b^3*d^2
*e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3)*x)/((a^3*b^4*d^4 - 4*a^4*b^3*d^3*e + 6*a^5*b
^2*d^2*e^2 - 4*a^6*b*d*e^3 + a^7*e^4 + (b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*
e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*x^3 + 3*(a*b^6*d^4 - 4*a^2*b^5*d^3*e + 6*a^
3*b^4*d^2*e^2 - 4*a^4*b^3*d*e^3 + a^5*b^2*e^4)*x^2 + 3*(a^2*b^5*d^4 - 4*a^3*b^4*
d^3*e + 6*a^4*b^3*d^2*e^2 - 4*a^5*b^2*d*e^3 + a^6*b*e^4)*x)*sqrt(e*x + d)), -1/2
4*(105*b^3*e^3*x^3 + 8*b^3*d^3 - 38*a*b^2*d^2*e + 87*a^2*b*d*e^2 + 48*a^3*e^3 +
35*(b^3*d*e^2 + 8*a*b^2*e^3)*x^2 - 105*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*
e^3*x + a^3*e^3)*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*(2*b^3*d^2*e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3
)*x)/((a^3*b^4*d^4 - 4*a^4*b^3*d^3*e + 6*a^5*b^2*d^2*e^2 - 4*a^6*b*d*e^3 + a^7*e
^4 + (b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^
4)*x^3 + 3*(a*b^6*d^4 - 4*a^2*b^5*d^3*e + 6*a^3*b^4*d^2*e^2 - 4*a^4*b^3*d*e^3 +
a^5*b^2*e^4)*x^2 + 3*(a^2*b^5*d^4 - 4*a^3*b^4*d^3*e + 6*a^4*b^3*d^2*e^2 - 4*a^5*
b^2*d*e^3 + a^6*b*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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.327219, size = 903, normalized size = 3.27 \[ \frac{35 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \, e^{3}}{{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{x e + d}} + \frac{57 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{3} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{3} + 87 \, \sqrt{x e + d} b^{3} d^{2} e^{3} + 136 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} e^{4} - 174 \, \sqrt{x e + d} a b^{2} d e^{4} + 87 \, \sqrt{x e + d} a^{2} b e^{5}}{24 \,{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

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

[Out]

35/8*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^4*d^4*sign(-(x*e + d
)*b*e + b*d*e - a*e^2) - 4*a*b^3*d^3*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 6*
a^2*b^2*d^2*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^3*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2) + a^4*e^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sqrt(-
b^2*d + a*b*e)) + 2*e^3/((b^4*d^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a*b^3
*d^3*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 6*a^2*b^2*d^2*e^2*sign(-(x*e + d)*
b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + a^4*
e^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sqrt(x*e + d)) + 1/24*(57*(x*e + d)^(5
/2)*b^3*e^3 - 136*(x*e + d)^(3/2)*b^3*d*e^3 + 87*sqrt(x*e + d)*b^3*d^2*e^3 + 136
*(x*e + d)^(3/2)*a*b^2*e^4 - 174*sqrt(x*e + d)*a*b^2*d*e^4 + 87*sqrt(x*e + d)*a^
2*b*e^5)/((b^4*d^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a*b^3*d^3*e*sign(-(x
*e + d)*b*e + b*d*e - a*e^2) + 6*a^2*b^2*d^2*e^2*sign(-(x*e + d)*b*e + b*d*e - a
*e^2) - 4*a^3*b*d*e^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + a^4*e^4*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2))*((x*e + d)*b - b*d + a*e)^3)