3.1827 \(\int \frac{(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)
Optimal. Leaf size=313 \[ -\frac{7 e^4 (-9 a B e-A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}}-\frac{7 e^3 \sqrt{d+e x} (-9 a B e-A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}-\frac{7 e^2 (d+e x)^{3/2} (-9 a B e-A b e+10 b B d)}{192 b^4 (a+b x)^2 (b d-a e)}-\frac{7 e (d+e x)^{5/2} (-9 a B e-A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e-A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
[Out]
(-7*e^3*(10*b*B*d - A*b*e - 9*a*B*e)*Sqrt[d + e*x])/(128*b^5*(b*d - a*e)*(a + b*
x)) - (7*e^2*(10*b*B*d - A*b*e - 9*a*B*e)*(d + e*x)^(3/2))/(192*b^4*(b*d - a*e)*
(a + b*x)^2) - (7*e*(10*b*B*d - A*b*e - 9*a*B*e)*(d + e*x)^(5/2))/(240*b^3*(b*d
- a*e)*(a + b*x)^3) - ((10*b*B*d - A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(40*b^2*(b*
d - a*e)*(a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(9/2))/(5*b*(b*d - a*e)*(a + b*x)
^5) - (7*e^4*(10*b*B*d - A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b
*d - a*e]])/(128*b^(11/2)*(b*d - a*e)^(3/2))
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Rubi [A] time = 0.589407, antiderivative size = 313, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152
\[ -\frac{7 e^4 (-9 a B e-A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}}-\frac{7 e^3 \sqrt{d+e x} (-9 a B e-A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}-\frac{7 e^2 (d+e x)^{3/2} (-9 a B e-A b e+10 b B d)}{192 b^4 (a+b x)^2 (b d-a e)}-\frac{7 e (d+e x)^{5/2} (-9 a B e-A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e-A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
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,x]
[Out]
(-7*e^3*(10*b*B*d - A*b*e - 9*a*B*e)*Sqrt[d + e*x])/(128*b^5*(b*d - a*e)*(a + b*
x)) - (7*e^2*(10*b*B*d - A*b*e - 9*a*B*e)*(d + e*x)^(3/2))/(192*b^4*(b*d - a*e)*
(a + b*x)^2) - (7*e*(10*b*B*d - A*b*e - 9*a*B*e)*(d + e*x)^(5/2))/(240*b^3*(b*d
- a*e)*(a + b*x)^3) - ((10*b*B*d - A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(40*b^2*(b*
d - a*e)*(a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(9/2))/(5*b*(b*d - a*e)*(a + b*x)
^5) - (7*e^4*(10*b*B*d - A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b
*d - a*e]])/(128*b^(11/2)*(b*d - a*e)^(3/2))
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Rubi in Sympy [A] time = 117.226, size = 292, normalized size = 0.93 \[ \frac{\left (d + e x\right )^{\frac{9}{2}} \left (A b - B a\right )}{5 b \left (a + b x\right )^{5} \left (a e - b d\right )} - \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b e + 9 B a e - 10 B b d\right )}{40 b^{2} \left (a + b x\right )^{4} \left (a e - b d\right )} - \frac{7 e \left (d + e x\right )^{\frac{5}{2}} \left (A b e + 9 B a e - 10 B b d\right )}{240 b^{3} \left (a + b x\right )^{3} \left (a e - b d\right )} - \frac{7 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (A b e + 9 B a e - 10 B b d\right )}{192 b^{4} \left (a + b x\right )^{2} \left (a e - b d\right )} - \frac{7 e^{3} \sqrt{d + e x} \left (A b e + 9 B a e - 10 B b d\right )}{128 b^{5} \left (a + b x\right ) \left (a e - b d\right )} + \frac{7 e^{4} \left (A b e + 9 B a e - 10 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{11}{2}} \left (a e - b d\right )^{\frac{3}{2}}} \]
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,x)
[Out]
(d + e*x)**(9/2)*(A*b - B*a)/(5*b*(a + b*x)**5*(a*e - b*d)) - (d + e*x)**(7/2)*(
A*b*e + 9*B*a*e - 10*B*b*d)/(40*b**2*(a + b*x)**4*(a*e - b*d)) - 7*e*(d + e*x)**
(5/2)*(A*b*e + 9*B*a*e - 10*B*b*d)/(240*b**3*(a + b*x)**3*(a*e - b*d)) - 7*e**2*
(d + e*x)**(3/2)*(A*b*e + 9*B*a*e - 10*B*b*d)/(192*b**4*(a + b*x)**2*(a*e - b*d)
) - 7*e**3*sqrt(d + e*x)*(A*b*e + 9*B*a*e - 10*B*b*d)/(128*b**5*(a + b*x)*(a*e -
b*d)) + 7*e**4*(A*b*e + 9*B*a*e - 10*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e
- b*d))/(128*b**(11/2)*(a*e - b*d)**(3/2))
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Mathematica [A] time = 0.888446, size = 255, normalized size = 0.81 \[ \frac{7 e^4 (9 a B e+A b e-10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} \left (15 e^3 (a+b x)^4 (-193 a B e+7 A b e+186 b B d)+10 e^2 (a+b x)^3 (b d-a e) (-447 a B e+121 A b e+326 b B d)+48 (a+b x) (b d-a e)^3 (-41 a B e+31 A b e+10 b B d)+8 e (a+b x)^2 (b d-a e)^2 (-513 a B e+263 A b e+250 b B d)+384 (A b-a B) (b d-a e)^4\right )}{1920 b^5 (a+b x)^5 (b d-a e)} \]
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,x]
[Out]
-(Sqrt[d + e*x]*(384*(A*b - a*B)*(b*d - a*e)^4 + 48*(b*d - a*e)^3*(10*b*B*d + 31
*A*b*e - 41*a*B*e)*(a + b*x) + 8*e*(b*d - a*e)^2*(250*b*B*d + 263*A*b*e - 513*a*
B*e)*(a + b*x)^2 + 10*e^2*(b*d - a*e)*(326*b*B*d + 121*A*b*e - 447*a*B*e)*(a + b
*x)^3 + 15*e^3*(186*b*B*d + 7*A*b*e - 193*a*B*e)*(a + b*x)^4))/(1920*b^5*(b*d -
a*e)*(a + b*x)^5) + (7*e^4*(-10*b*B*d + A*b*e + 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d
+ e*x])/Sqrt[b*d - a*e]])/(128*b^(11/2)*(b*d - a*e)^(3/2))
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Maple [B] time = 0.036, size = 959, normalized size = 3.1 \[ \text{result too large to display} \]
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,x)
[Out]
7/128*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(9/2)*A+245/96*e^4/(b*e*x+a*e)^5/b*(e*
x+d)^(3/2)*B*d^3-49/192*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*A*d^2-49/192*e^7/(b*e*
x+a*e)^5/b^3*(e*x+d)^(3/2)*A*a^2+7/128*e^5/b^4/(a*e-b*d)/(b*(a*e-b*d))^(1/2)*arc
tan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A+395/96*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(7
/2)*B*d-14/3*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*B*d^2-35/64*e^4/(b*e*x+a*e)^5/b*(
e*x+d)^(1/2)*B*d^4+93/64*e^4/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(9/2)*B*d-63/128*e^
8/(b*e*x+a*e)^5/b^5*(e*x+d)^(1/2)*B*a^4-147/64*e^7/(b*e*x+a*e)^5/b^4*(e*x+d)^(3/
2)*B*a^3-7/128*e^8/(b*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*A*a^3+7/128*e^5/(b*e*x+a*e)^5
/b*(e*x+d)^(1/2)*A*d^3+7/15*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*A*d-21/5*e^6/(b*e*
x+a*e)^5/b^3*(e*x+d)^(5/2)*a^2*B-7/15*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(5/2)*A*a-23
7/64*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(7/2)*a*B+133/15*e^5/(b*e*x+a*e)^5/b^2*(e*x+d
)^(5/2)*B*d*a+49/96*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*A*a*d-35/64*e^4/b^4/(a*e
-b*d)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*d+21/128
*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*A*d*a^2-21/128*e^6/(b*e*x+a*e)^5/b^2*(e*x+d
)^(1/2)*A*a*d^2+259/128*e^7/(b*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*B*a^3*d-399/128*e^6/
(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*B*d^2*a^2+273/128*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^
(1/2)*B*a*d^3-193/128*e^5/(b*e*x+a*e)^5/b/(a*e-b*d)*(e*x+d)^(9/2)*a*B+63/128*e^5
/b^5/(a*e-b*d)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a
*B-1421/192*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*B*a*d^2+343/48*e^6/(b*e*x+a*e)^5
/b^3*(e*x+d)^(3/2)*B*a^2*d-79/192*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(7/2)*A
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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)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.319133, 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)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
[-1/3840*(2*(96*(B*a*b^4 + 4*A*b^5)*d^4 + 16*(8*B*a^2*b^3 - 3*A*a*b^4)*d^3*e + 2
8*(7*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e^2 + 70*(6*B*a^4*b - A*a^3*b^2)*d*e^3 - 105*(
9*B*a^5 + A*a^4*b)*e^4 + 15*(186*B*b^5*d*e^3 - (193*B*a*b^4 - 7*A*b^5)*e^4)*x^4
+ 10*(326*B*b^5*d^2*e^2 + (343*B*a*b^4 + 121*A*b^5)*d*e^3 - 79*(9*B*a^2*b^3 + A*
a*b^4)*e^4)*x^3 + 2*(1000*B*b^5*d^3*e + 2*(419*B*a*b^4 + 526*A*b^5)*d^2*e^2 + (1
879*B*a^2*b^3 - 289*A*a*b^4)*d*e^3 - 448*(9*B*a^3*b^2 + A*a^2*b^3)*e^4)*x^2 + 2*
(240*B*b^5*d^4 + 8*(37*B*a*b^4 + 93*A*b^5)*d^3*e + 2*(229*B*a^2*b^3 - 64*A*a*b^4
)*d^2*e^2 + 7*(143*B*a^3*b^2 - 23*A*a^2*b^3)*d*e^3 - 245*(9*B*a^4*b + A*a^3*b^2)
*e^4)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) - 105*(10*B*a^5*b*d*e^4 - (9*B*a^6 +
A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (9*B*a*b^5 + A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*
e^4 - (9*B*a^2*b^4 + A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (9*B*a^3*b^3 +
A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (9*B*a^4*b^2 + A*a^3*b^3)*e^5)*x
^2 + 5*(10*B*a^4*b^2*d*e^4 - (9*B*a^5*b + A*a^4*b^2)*e^5)*x)*log((sqrt(b^2*d - a
*b*e)*(b*e*x + 2*b*d - a*e) - 2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((a^5
*b^6*d - a^6*b^5*e + (b^11*d - a*b^10*e)*x^5 + 5*(a*b^10*d - a^2*b^9*e)*x^4 + 10
*(a^2*b^9*d - a^3*b^8*e)*x^3 + 10*(a^3*b^8*d - a^4*b^7*e)*x^2 + 5*(a^4*b^7*d - a
^5*b^6*e)*x)*sqrt(b^2*d - a*b*e)), -1/1920*((96*(B*a*b^4 + 4*A*b^5)*d^4 + 16*(8*
B*a^2*b^3 - 3*A*a*b^4)*d^3*e + 28*(7*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e^2 + 70*(6*B*
a^4*b - A*a^3*b^2)*d*e^3 - 105*(9*B*a^5 + A*a^4*b)*e^4 + 15*(186*B*b^5*d*e^3 - (
193*B*a*b^4 - 7*A*b^5)*e^4)*x^4 + 10*(326*B*b^5*d^2*e^2 + (343*B*a*b^4 + 121*A*b
^5)*d*e^3 - 79*(9*B*a^2*b^3 + A*a*b^4)*e^4)*x^3 + 2*(1000*B*b^5*d^3*e + 2*(419*B
*a*b^4 + 526*A*b^5)*d^2*e^2 + (1879*B*a^2*b^3 - 289*A*a*b^4)*d*e^3 - 448*(9*B*a^
3*b^2 + A*a^2*b^3)*e^4)*x^2 + 2*(240*B*b^5*d^4 + 8*(37*B*a*b^4 + 93*A*b^5)*d^3*e
+ 2*(229*B*a^2*b^3 - 64*A*a*b^4)*d^2*e^2 + 7*(143*B*a^3*b^2 - 23*A*a^2*b^3)*d*e
^3 - 245*(9*B*a^4*b + A*a^3*b^2)*e^4)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) + 10
5*(10*B*a^5*b*d*e^4 - (9*B*a^6 + A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (9*B*a*b^5 + A
*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (9*B*a^2*b^4 + A*a*b^5)*e^5)*x^4 + 10*(10
*B*a^2*b^4*d*e^4 - (9*B*a^3*b^3 + A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 -
(9*B*a^4*b^2 + A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (9*B*a^5*b + A*a^4
*b^2)*e^5)*x)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a^5*b
^6*d - a^6*b^5*e + (b^11*d - a*b^10*e)*x^5 + 5*(a*b^10*d - a^2*b^9*e)*x^4 + 10*(
a^2*b^9*d - a^3*b^8*e)*x^3 + 10*(a^3*b^8*d - a^4*b^7*e)*x^2 + 5*(a^4*b^7*d - a^5
*b^6*e)*x)*sqrt(-b^2*d + a*b*e))]
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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,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.319322, size = 1, normalized size = 0. \[ \mathit{Done} \]
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,x, algorithm="giac")
[Out]
Done