3.2144 \(\int \frac{a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=328 \[ -\frac{105 b e^3 (a+b x)}{8 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}-\frac{35 e^3 (a+b x)}{8 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac{21 e^2}{8 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac{3 e}{4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}+\frac{105 b^{3/2} e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \]

[Out]

(-21*e^2)/(8*(b*d - a*e)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(3
*(b*d - a*e)*(a + b*x)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e)/
(4*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*
e^3*(a + b*x))/(8*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(105*b*e^3*(a + b*x))/(8*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (105*b^(3/2)*e^3*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(8*(b*d - a*e)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.568091, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{105 b e^3 (a+b x)}{8 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}-\frac{35 e^3 (a+b x)}{8 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac{21 e^2}{8 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac{3 e}{4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}+\frac{105 b^{3/2} e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-21*e^2)/(8*(b*d - a*e)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(3
*(b*d - a*e)*(a + b*x)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e)/
(4*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*
e^3*(a + b*x))/(8*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(105*b*e^3*(a + b*x))/(8*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (105*b^(3/2)*e^3*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(8*(b*d - a*e)^(11/2)*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**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.191, size = 186, normalized size = 0.57 \[ \frac{(a+b x) \left (\frac{105 b^{3/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{11/2}}+\frac{\sqrt{d+e x} \left (\frac{34 b^2 e (b d-a e)}{(a+b x)^2}-\frac{8 b^2 (b d-a e)^2}{(a+b x)^3}-\frac{123 b^2 e^2}{a+b x}+\frac{16 e^3 (a e-b d)}{(d+e x)^2}-\frac{192 b e^3}{d+e x}\right )}{3 (b d-a e)^5}\right )}{8 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*((Sqrt[d + e*x]*((-8*b^2*(b*d - a*e)^2)/(a + b*x)^3 + (34*b^2*e*(b*d
- a*e))/(a + b*x)^2 - (123*b^2*e^2)/(a + b*x) + (16*e^3*(-(b*d) + a*e))/(d + e*x
)^2 - (192*b*e^3)/(d + e*x)))/(3*(b*d - a*e)^5) + (105*b^(3/2)*e^3*ArcTanh[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(11/2)))/(8*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.035, size = 563, normalized size = 1.7 \[{\frac{ \left ( bx+a \right ) ^{2}}{24\, \left ( ae-bd \right ) ^{5}} \left ( 315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}{x}^{3}{b}^{5}{e}^{3}+945\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}{x}^{2}a{b}^{4}{e}^{3}+315\,\sqrt{b \left ( ae-bd \right ) }{x}^{4}{b}^{4}{e}^{4}+945\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}x{a}^{2}{b}^{3}{e}^{3}+840\,\sqrt{b \left ( ae-bd \right ) }{x}^{3}a{b}^{3}{e}^{4}+420\,\sqrt{b \left ( ae-bd \right ) }{x}^{3}{b}^{4}d{e}^{3}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}{a}^{3}{b}^{2}{e}^{3}+693\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{a}^{2}{b}^{2}{e}^{4}+1134\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}a{b}^{3}d{e}^{3}+63\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{4}{d}^{2}{e}^{2}+144\,\sqrt{b \left ( ae-bd \right ) }x{a}^{3}b{e}^{4}+954\,\sqrt{b \left ( ae-bd \right ) }x{a}^{2}{b}^{2}d{e}^{3}+180\,\sqrt{b \left ( ae-bd \right ) }xa{b}^{3}{d}^{2}{e}^{2}-18\,\sqrt{b \left ( ae-bd \right ) }x{b}^{4}{d}^{3}e-16\,\sqrt{b \left ( ae-bd \right ) }{a}^{4}{e}^{4}+208\,\sqrt{b \left ( ae-bd \right ) }{a}^{3}bd{e}^{3}+165\,\sqrt{b \left ( ae-bd \right ) }{a}^{2}{b}^{2}{d}^{2}{e}^{2}-50\,\sqrt{b \left ( ae-bd \right ) }a{b}^{3}{d}^{3}e+8\,\sqrt{b \left ( ae-bd \right ) }{b}^{4}{d}^{4} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( ex+d \right ) ^{-{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^3*b^5*e^3+
945*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^2*a*b^4*e^3+315*
(b*(a*e-b*d))^(1/2)*x^4*b^4*e^4+945*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*
(e*x+d)^(3/2)*x*a^2*b^3*e^3+840*(b*(a*e-b*d))^(1/2)*x^3*a*b^3*e^4+420*(b*(a*e-b*
d))^(1/2)*x^3*b^4*d*e^3+315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^
(3/2)*a^3*b^2*e^3+693*(b*(a*e-b*d))^(1/2)*x^2*a^2*b^2*e^4+1134*(b*(a*e-b*d))^(1/
2)*x^2*a*b^3*d*e^3+63*(b*(a*e-b*d))^(1/2)*x^2*b^4*d^2*e^2+144*(b*(a*e-b*d))^(1/2
)*x*a^3*b*e^4+954*(b*(a*e-b*d))^(1/2)*x*a^2*b^2*d*e^3+180*(b*(a*e-b*d))^(1/2)*x*
a*b^3*d^2*e^2-18*(b*(a*e-b*d))^(1/2)*x*b^4*d^3*e-16*(b*(a*e-b*d))^(1/2)*a^4*e^4+
208*(b*(a*e-b*d))^(1/2)*a^3*b*d*e^3+165*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2-50*(
b*(a*e-b*d))^(1/2)*a*b^3*d^3*e+8*(b*(a*e-b*d))^(1/2)*b^4*d^4)*(b*x+a)^2/(b*(a*e-
b*d))^(1/2)/(e*x+d)^(3/2)/(a*e-b*d)^5/((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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.327708, 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)^(5/2)),x, algorithm="fricas")

[Out]

[-1/48*(630*b^4*e^4*x^4 + 16*b^4*d^4 - 100*a*b^3*d^3*e + 330*a^2*b^2*d^2*e^2 + 4
16*a^3*b*d*e^3 - 32*a^4*e^4 + 840*(b^4*d*e^3 + 2*a*b^3*e^4)*x^3 + 126*(b^4*d^2*e
^2 + 18*a*b^3*d*e^3 + 11*a^2*b^2*e^4)*x^2 + 315*(b^4*e^4*x^4 + a^3*b*d*e^3 + (b^
4*d*e^3 + 3*a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + (3*a^2*b^2*d*e^
3 + a^3*b*e^4)*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)) - 36*(b^4*d^3*e - 10*
a*b^3*d^2*e^2 - 53*a^2*b^2*d*e^3 - 8*a^3*b*e^4)*x)/((a^3*b^5*d^6 - 5*a^4*b^4*d^5
*e + 10*a^5*b^3*d^4*e^2 - 10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b^
8*d^5*e - 5*a*b^7*d^4*e^2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^4 + 5*a^4*b^4*
d*e^5 - a^5*b^3*e^6)*x^4 + (b^8*d^6 - 2*a*b^7*d^5*e - 5*a^2*b^6*d^4*e^2 + 20*a^3
*b^5*d^3*e^3 - 25*a^4*b^4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e^6)*x^3 + 3*(a
*b^7*d^6 - 4*a^2*b^6*d^5*e + 5*a^3*b^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4 + 4*a^6*b^2*d
*e^5 - a^7*b*e^6)*x^2 + (3*a^2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4*b^4*d^4*e^2 -
20*a^5*b^3*d^3*e^3 + 5*a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a^8*e^6)*x)*sqrt(e*x +
d)), -1/24*(315*b^4*e^4*x^4 + 8*b^4*d^4 - 50*a*b^3*d^3*e + 165*a^2*b^2*d^2*e^2
+ 208*a^3*b*d*e^3 - 16*a^4*e^4 + 420*(b^4*d*e^3 + 2*a*b^3*e^4)*x^3 + 63*(b^4*d^2
*e^2 + 18*a*b^3*d*e^3 + 11*a^2*b^2*e^4)*x^2 - 315*(b^4*e^4*x^4 + a^3*b*d*e^3 + (
b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + (3*a^2*b^2*d*
e^3 + a^3*b*e^4)*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)) - 18*(b^4*d^3*e - 10*a*b^3*d^2*e^2 - 53*a^2*b
^2*d*e^3 - 8*a^3*b*e^4)*x)/((a^3*b^5*d^6 - 5*a^4*b^4*d^5*e + 10*a^5*b^3*d^4*e^2
- 10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b^8*d^5*e - 5*a*b^7*d^4*e^
2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^4 + 5*a^4*b^4*d*e^5 - a^5*b^3*e^6)*x^4
+ (b^8*d^6 - 2*a*b^7*d^5*e - 5*a^2*b^6*d^4*e^2 + 20*a^3*b^5*d^3*e^3 - 25*a^4*b^
4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e^6)*x^3 + 3*(a*b^7*d^6 - 4*a^2*b^6*d^5
*e + 5*a^3*b^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4 + 4*a^6*b^2*d*e^5 - a^7*b*e^6)*x^2 +
(3*a^2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4*b^4*d^4*e^2 - 20*a^5*b^3*d^3*e^3 + 5*
a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a^8*e^6)*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.330517, size = 949, normalized size = 2.89 \[ \text{result too large to display} \]

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)^(5/2)),x, algorithm="giac")

[Out]

105/8*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^5*d^5*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2) - 5*a*b^4*d^4*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) +
10*a^2*b^3*d^3*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 10*a^3*b^2*d^2*e^3*si
gn(-(x*e + d)*b*e + b*d*e - a*e^2) + 5*a^4*b*d*e^4*sign(-(x*e + d)*b*e + b*d*e -
a*e^2) - a^5*e^5*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sqrt(-b^2*d + a*b*e)) +
1/24*(315*(x*e + d)^4*b^4*e^3 - 840*(x*e + d)^3*b^4*d*e^3 + 693*(x*e + d)^2*b^4*
d^2*e^3 - 144*(x*e + d)*b^4*d^3*e^3 - 16*b^4*d^4*e^3 + 840*(x*e + d)^3*a*b^3*e^4
- 1386*(x*e + d)^2*a*b^3*d*e^4 + 432*(x*e + d)*a*b^3*d^2*e^4 + 64*a*b^3*d^3*e^4
+ 693*(x*e + d)^2*a^2*b^2*e^5 - 432*(x*e + d)*a^2*b^2*d*e^5 - 96*a^2*b^2*d^2*e^
5 + 144*(x*e + d)*a^3*b*e^6 + 64*a^3*b*d*e^6 - 16*a^4*e^7)/((b^5*d^5*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2) - 5*a*b^4*d^4*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) +
10*a^2*b^3*d^3*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 10*a^3*b^2*d^2*e^3*si
gn(-(x*e + d)*b*e + b*d*e - a*e^2) + 5*a^4*b*d*e^4*sign(-(x*e + d)*b*e + b*d*e -
a*e^2) - a^5*e^5*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*((x*e + d)^(3/2)*b - sqr
t(x*e + d)*b*d + sqrt(x*e + d)*a*e)^3)