3.1708 \(\int \frac{1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=329 \[ \frac{63 b^2 e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}+\frac{21 b e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac{63 e^2 (a+b x)}{20 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac{9 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{63 b^{5/2} e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \]
[Out]
(9*e)/(4*(b*d - a*e)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(2*(b*
d - a*e)*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (63*e^2*(a +
b*x))/(20*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (21*b*
e^2*(a + b*x))/(4*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(63*b^2*e^2*(a + b*x))/(4*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) - (63*b^(5/2)*e^2*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(4*(b*d - a*e)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
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Rubi [A] time = 0.533324, antiderivative size = 329, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167
\[ \frac{63 b^2 e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}+\frac{21 b e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac{63 e^2 (a+b x)}{20 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac{9 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{63 b^{5/2} e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
(9*e)/(4*(b*d - a*e)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(2*(b*
d - a*e)*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (63*e^2*(a +
b*x))/(20*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (21*b*
e^2*(a + b*x))/(4*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(63*b^2*e^2*(a + b*x))/(4*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) - (63*b^(5/2)*e^2*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(4*(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(1/(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 = 1.38737, size = 188, normalized size = 0.57 \[ \frac{(a+b x)^3 \left (\frac{\sqrt{d+e x} \left (-\frac{10 b^3 (b d-a e)}{(a+b x)^2}+\frac{75 b^3 e}{a+b x}+\frac{40 b e^2 (b d-a e)}{(d+e x)^2}+\frac{8 e^2 (b d-a e)^2}{(d+e x)^3}+\frac{240 b^2 e^2}{d+e x}\right )}{5 (b d-a e)^5}-\frac{63 b^{5/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{11/2}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
((a + b*x)^3*((Sqrt[d + e*x]*((-10*b^3*(b*d - a*e))/(a + b*x)^2 + (75*b^3*e)/(a
+ b*x) + (8*e^2*(b*d - a*e)^2)/(d + e*x)^3 + (40*b*e^2*(b*d - a*e))/(d + e*x)^2
+ (240*b^2*e^2)/(d + e*x)))/(5*(b*d - a*e)^5) - (63*b^(5/2)*e^2*ArcTanh[(Sqrt[b]
*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(11/2)))/(4*((a + b*x)^2)^(3/2))
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Maple [B] time = 0.032, size = 518, normalized size = 1.6 \[ -{\frac{bx+a}{20\, \left ( ae-bd \right ) ^{5}} \left ( 315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{5/2}{x}^{2}{b}^{5}{e}^{2}+630\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{5/2}xa{b}^{4}{e}^{2}+315\,\sqrt{b \left ( ae-bd \right ) }{x}^{4}{b}^{4}{e}^{4}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{5/2}{a}^{2}{b}^{3}{e}^{2}+525\,\sqrt{b \left ( ae-bd \right ) }{x}^{3}a{b}^{3}{e}^{4}+735\,\sqrt{b \left ( ae-bd \right ) }{x}^{3}{b}^{4}d{e}^{3}+168\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{a}^{2}{b}^{2}{e}^{4}+1239\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}a{b}^{3}d{e}^{3}+483\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{4}{d}^{2}{e}^{2}-24\,\sqrt{b \left ( ae-bd \right ) }x{a}^{3}b{e}^{4}+408\,\sqrt{b \left ( ae-bd \right ) }x{a}^{2}{b}^{2}d{e}^{3}+831\,\sqrt{b \left ( ae-bd \right ) }xa{b}^{3}{d}^{2}{e}^{2}+45\,\sqrt{b \left ( ae-bd \right ) }x{b}^{4}{d}^{3}e+8\,\sqrt{b \left ( ae-bd \right ) }{a}^{4}{e}^{4}-56\,\sqrt{b \left ( ae-bd \right ) }{a}^{3}bd{e}^{3}+288\,\sqrt{b \left ( ae-bd \right ) }{a}^{2}{b}^{2}{d}^{2}{e}^{2}+85\,\sqrt{b \left ( ae-bd \right ) }a{b}^{3}{d}^{3}e-10\,\sqrt{b \left ( ae-bd \right ) }{b}^{4}{d}^{4} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( ex+d \right ) ^{-{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
-1/20*(315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*x^2*b^5*e^2
+630*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*x*a*b^4*e^2+315*(
b*(a*e-b*d))^(1/2)*x^4*b^4*e^4+315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(
e*x+d)^(5/2)*a^2*b^3*e^2+525*(b*(a*e-b*d))^(1/2)*x^3*a*b^3*e^4+735*(b*(a*e-b*d))
^(1/2)*x^3*b^4*d*e^3+168*(b*(a*e-b*d))^(1/2)*x^2*a^2*b^2*e^4+1239*(b*(a*e-b*d))^
(1/2)*x^2*a*b^3*d*e^3+483*(b*(a*e-b*d))^(1/2)*x^2*b^4*d^2*e^2-24*(b*(a*e-b*d))^(
1/2)*x*a^3*b*e^4+408*(b*(a*e-b*d))^(1/2)*x*a^2*b^2*d*e^3+831*(b*(a*e-b*d))^(1/2)
*x*a*b^3*d^2*e^2+45*(b*(a*e-b*d))^(1/2)*x*b^4*d^3*e+8*(b*(a*e-b*d))^(1/2)*a^4*e^
4-56*(b*(a*e-b*d))^(1/2)*a^3*b*d*e^3+288*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2+85*
(b*(a*e-b*d))^(1/2)*a*b^3*d^3*e-10*(b*(a*e-b*d))^(1/2)*b^4*d^4)*(b*x+a)/(b*(a*e-
b*d))^(1/2)/(e*x+d)^(5/2)/(a*e-b*d)^5/((b*x+a)^2)^(3/2)
_______________________________________________________________________________________
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(1/((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 = 0.23759, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(7/2)),x, algorithm="fricas")
[Out]
[1/40*(630*b^4*e^4*x^4 - 20*b^4*d^4 + 170*a*b^3*d^3*e + 576*a^2*b^2*d^2*e^2 - 11
2*a^3*b*d*e^3 + 16*a^4*e^4 + 210*(7*b^4*d*e^3 + 5*a*b^3*e^4)*x^3 + 42*(23*b^4*d^
2*e^2 + 59*a*b^3*d*e^3 + 8*a^2*b^2*e^4)*x^2 - 315*(b^4*e^4*x^4 + a^2*b^2*d^2*e^2
+ 2*(b^4*d*e^3 + a*b^3*e^4)*x^3 + (b^4*d^2*e^2 + 4*a*b^3*d*e^3 + a^2*b^2*e^4)*x
^2 + 2*(a*b^3*d^2*e^2 + a^2*b^2*d*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
)) + 6*(15*b^4*d^3*e + 277*a*b^3*d^2*e^2 + 136*a^2*b^2*d*e^3 - 8*a^3*b*e^4)*x)/(
(a^2*b^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6
*b*d^3*e^4 - a^7*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 -
10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^
6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*e^7)
*x^3 + (b^7*d^7 - a*b^6*d^6*e - 9*a^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*
b^3*d^3*e^4 + 9*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 - a^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*
a^2*b^5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 + 4*a^6*b*d^2*e^5 - a^7*d*
e^6)*x)*sqrt(e*x + d)), 1/20*(315*b^4*e^4*x^4 - 10*b^4*d^4 + 85*a*b^3*d^3*e + 28
8*a^2*b^2*d^2*e^2 - 56*a^3*b*d*e^3 + 8*a^4*e^4 + 105*(7*b^4*d*e^3 + 5*a*b^3*e^4)
*x^3 + 21*(23*b^4*d^2*e^2 + 59*a*b^3*d*e^3 + 8*a^2*b^2*e^4)*x^2 - 315*(b^4*e^4*x
^4 + a^2*b^2*d^2*e^2 + 2*(b^4*d*e^3 + a*b^3*e^4)*x^3 + (b^4*d^2*e^2 + 4*a*b^3*d*
e^3 + a^2*b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 + a^2*b^2*d*e^3)*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)) + 3
*(15*b^4*d^3*e + 277*a*b^3*d^2*e^2 + 136*a^2*b^2*d*e^3 - 8*a^3*b*e^4)*x)/((a^2*b
^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3
*e^4 - a^7*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^
3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^6*d^5*
e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*e^7)*x^3 +
(b^7*d^7 - a*b^6*d^6*e - 9*a^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*b^3*d^
3*e^4 + 9*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 - a^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^
5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 + 4*a^6*b*d^2*e^5 - a^7*d*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(1/(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 = 0.262178, size = 1071, normalized size = 3.26 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(7/2)),x, algorithm="giac")
[Out]
-63/4*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^2/((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)) -
1/4*(15*(x*e + d)^(3/2)*b^4*e^2 - 17*sqrt(x*e + d)*b^4*d*e^2 + 17*sqrt(x*e + d)*
a*b^3*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*sign(-(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)*b - b*d + a*e)^2) - 2/5*(30*(x*e + d)^2*b^2*e^2 + 5*(x*e + d)
*b^2*d*e^2 + b^2*d^2*e^2 - 5*(x*e + d)*a*b*e^3 - 2*a*b*d*e^3 + a^2*e^4)/((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*sign(-(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)^(
5/2))