3.2081 \(\int \frac{a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=167 \[ -\frac{35 b^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}}+\frac{35 b e^2}{4 \sqrt{d+e x} (b d-a e)^4}+\frac{35 e^2}{12 (d+e x)^{3/2} (b d-a e)^3}+\frac{7 e}{4 (a+b x) (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.278098, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{35 b^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}}+\frac{35 b e^2}{4 \sqrt{d+e x} (b d-a e)^4}+\frac{35 e^2}{12 (d+e x)^{3/2} (b d-a e)^3}+\frac{7 e}{4 (a+b x) (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 73.6335, size = 148, normalized size = 0.89 \[ \frac{35 b^{\frac{3}{2}} e^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 \left (a e - b d\right )^{\frac{9}{2}}} + \frac{35 b e^{2}}{4 \sqrt{d + e x} \left (a e - b d\right )^{4}} - \frac{35 e^{2}}{12 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} + \frac{7 e}{4 \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{1}{2 \left (a + b x\right )^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35*b**(3/2)*e**2*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(4*(a*e - b*d)**(9/
2)) + 35*b*e**2/(4*sqrt(d + e*x)*(a*e - b*d)**4) - 35*e**2/(12*(d + e*x)**(3/2)*
(a*e - b*d)**3) + 7*e/(4*(a + b*x)*(d + e*x)**(3/2)*(a*e - b*d)**2) + 1/(2*(a +
b*x)**2*(d + e*x)**(3/2)*(a*e - b*d))

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Mathematica [A]  time = 0.553318, size = 143, normalized size = 0.86 \[ \frac{\sqrt{d+e x} \left (-\frac{6 b^2 (b d-a e)}{(a+b x)^2}+\frac{33 b^2 e}{a+b x}+\frac{8 e^2 (b d-a e)}{(d+e x)^2}+\frac{72 b e^2}{d+e x}\right )}{12 (b d-a e)^4}-\frac{35 b^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((-6*b^2*(b*d - a*e))/(a + b*x)^2 + (33*b^2*e)/(a + b*x) + (8*e^2
*(b*d - a*e))/(d + e*x)^2 + (72*b*e^2)/(d + e*x)))/(12*(b*d - a*e)^4) - (35*b^(3
/2)*e^2*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(9/2))

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Maple [A]  time = 0.033, size = 206, normalized size = 1.2 \[ -{\frac{2\,{e}^{2}}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+6\,{\frac{b{e}^{2}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}+{\frac{11\,{b}^{3}{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{13\,a{b}^{2}{e}^{3}}{4\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{13\,{b}^{3}d{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,{b}^{2}{e}^{2}}{4\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*e^2/(a*e-b*d)^3/(e*x+d)^(3/2)+6*e^2/(a*e-b*d)^4*b/(e*x+d)^(1/2)+11/4*e^2*b^
3/(a*e-b*d)^4/(b*e*x+a*e)^2*(e*x+d)^(3/2)+13/4*e^3*b^2/(a*e-b*d)^4/(b*e*x+a*e)^2
*(e*x+d)^(1/2)*a-13/4*e^2*b^3/(a*e-b*d)^4/(b*e*x+a*e)^2*(e*x+d)^(1/2)*d+35/4*e^2
*b^2/(a*e-b*d)^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(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)/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30471, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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)^2*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/24*(210*b^3*e^3*x^3 - 12*b^3*d^3 + 78*a*b^2*d^2*e + 160*a^2*b*d*e^2 - 16*a^3*
e^3 + 70*(4*b^3*d*e^2 + 5*a*b^2*e^3)*x^2 + 105*(b^3*e^3*x^3 + a^2*b*d*e^2 + (b^3
*d*e^2 + 2*a*b^2*e^3)*x^2 + (2*a*b^2*d*e^2 + a^2*b*e^3)*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*(3*b^3*d^2*e + 34*a*b^2*d*e^2 + 8*a^2*b*e^3)*x)/((a^2*b
^4*d^5 - 4*a^3*b^3*d^4*e + 6*a^4*b^2*d^3*e^2 - 4*a^5*b*d^2*e^3 + a^6*d*e^4 + (b^
6*d^4*e - 4*a*b^5*d^3*e^2 + 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 + a^4*b^2*e^5)*x
^3 + (b^6*d^5 - 2*a*b^5*d^4*e - 2*a^2*b^4*d^3*e^2 + 8*a^3*b^3*d^2*e^3 - 7*a^4*b^
2*d*e^4 + 2*a^5*b*e^5)*x^2 + (2*a*b^5*d^5 - 7*a^2*b^4*d^4*e + 8*a^3*b^3*d^3*e^2
- 2*a^4*b^2*d^2*e^3 - 2*a^5*b*d*e^4 + a^6*e^5)*x)*sqrt(e*x + d)), 1/12*(105*b^3*
e^3*x^3 - 6*b^3*d^3 + 39*a*b^2*d^2*e + 80*a^2*b*d*e^2 - 8*a^3*e^3 + 35*(4*b^3*d*
e^2 + 5*a*b^2*e^3)*x^2 - 105*(b^3*e^3*x^3 + a^2*b*d*e^2 + (b^3*d*e^2 + 2*a*b^2*e
^3)*x^2 + (2*a*b^2*d*e^2 + a^2*b*e^3)*x)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))*arct
an(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)*b)) + 7*(3*b^3*d^2*e + 34*a*
b^2*d*e^2 + 8*a^2*b*e^3)*x)/((a^2*b^4*d^5 - 4*a^3*b^3*d^4*e + 6*a^4*b^2*d^3*e^2
- 4*a^5*b*d^2*e^3 + a^6*d*e^4 + (b^6*d^4*e - 4*a*b^5*d^3*e^2 + 6*a^2*b^4*d^2*e^3
- 4*a^3*b^3*d*e^4 + a^4*b^2*e^5)*x^3 + (b^6*d^5 - 2*a*b^5*d^4*e - 2*a^2*b^4*d^3
*e^2 + 8*a^3*b^3*d^2*e^3 - 7*a^4*b^2*d*e^4 + 2*a^5*b*e^5)*x^2 + (2*a*b^5*d^5 - 7
*a^2*b^4*d^4*e + 8*a^3*b^3*d^3*e^2 - 2*a^4*b^2*d^2*e^3 - 2*a^5*b*d*e^4 + a^6*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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284821, size = 398, normalized size = 2.38 \[ \frac{35 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \,{\left (9 \,{\left (x e + d\right )} b e^{2} + b d e^{2} - a e^{3}\right )}}{3 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} + \frac{11 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{2} - 13 \, \sqrt{x e + d} b^{3} d e^{2} + 13 \, \sqrt{x e + d} a b^{2} e^{3}}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

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)^2*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

35/4*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^2/((b^4*d^4 - 4*a*b^3*d^
3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(-b^2*d + a*b*e)) + 2/3*(
9*(x*e + d)*b*e^2 + b*d*e^2 - a*e^3)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e
^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(x*e + d)^(3/2)) + 1/4*(11*(x*e + d)^(3/2)*b^3*e^2
- 13*sqrt(x*e + d)*b^3*d*e^2 + 13*sqrt(x*e + d)*a*b^2*e^3)/((b^4*d^4 - 4*a*b^3*
d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*((x*e + d)*b - b*d + a*e)^2
)