3.1808 \(\int \frac{(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=256 \[ -\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{\sqrt{d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}+\frac{(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac{(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac{(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

[Out]

((b*d - a*e)^2*(2*b*B*d + 7*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/b^5 + ((b*d - a*e)*(
2*b*B*d + 7*A*b*e - 9*a*B*e)*(d + e*x)^(3/2))/(3*b^4) + ((2*b*B*d + 7*A*b*e - 9*
a*B*e)*(d + e*x)^(5/2))/(5*b^3) + ((2*b*B*d + 7*A*b*e - 9*a*B*e)*(d + e*x)^(7/2)
)/(7*b^2*(b*d - a*e)) - ((A*b - a*B)*(d + e*x)^(9/2))/(b*(b*d - a*e)*(a + b*x))
- ((b*d - a*e)^(5/2)*(2*b*B*d + 7*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[b*d - a*e]])/b^(11/2)

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Rubi [A]  time = 0.69469, antiderivative size = 256, 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 \[ -\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{\sqrt{d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}+\frac{(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac{(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac{(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

((b*d - a*e)^2*(2*b*B*d + 7*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/b^5 + ((b*d - a*e)*(
2*b*B*d + 7*A*b*e - 9*a*B*e)*(d + e*x)^(3/2))/(3*b^4) + ((2*b*B*d + 7*A*b*e - 9*
a*B*e)*(d + e*x)^(5/2))/(5*b^3) + ((2*b*B*d + 7*A*b*e - 9*a*B*e)*(d + e*x)^(7/2)
)/(7*b^2*(b*d - a*e)) - ((A*b - a*B)*(d + e*x)^(9/2))/(b*(b*d - a*e)*(a + b*x))
- ((b*d - a*e)^(5/2)*(2*b*B*d + 7*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[b*d - a*e]])/b^(11/2)

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Rubi in Sympy [A]  time = 101.264, size = 245, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{\frac{9}{2}} \left (A b - B a\right )}{b \left (a + b x\right ) \left (a e - b d\right )} - \frac{\left (d + e x\right )^{\frac{7}{2}} \left (7 A b e - 9 B a e + 2 B b d\right )}{7 b^{2} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (7 A b e - 9 B a e + 2 B b d\right )}{5 b^{3}} - \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (7 A b e - 9 B a e + 2 B b d\right )}{3 b^{4}} + \frac{\sqrt{d + e x} \left (a e - b d\right )^{2} \left (7 A b e - 9 B a e + 2 B b d\right )}{b^{5}} - \frac{\left (a e - b d\right )^{\frac{5}{2}} \left (7 A b e - 9 B a e + 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{11}{2}}} \]

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),x)

[Out]

(d + e*x)**(9/2)*(A*b - B*a)/(b*(a + b*x)*(a*e - b*d)) - (d + e*x)**(7/2)*(7*A*b
*e - 9*B*a*e + 2*B*b*d)/(7*b**2*(a*e - b*d)) + (d + e*x)**(5/2)*(7*A*b*e - 9*B*a
*e + 2*B*b*d)/(5*b**3) - (d + e*x)**(3/2)*(a*e - b*d)*(7*A*b*e - 9*B*a*e + 2*B*b
*d)/(3*b**4) + sqrt(d + e*x)*(a*e - b*d)**2*(7*A*b*e - 9*B*a*e + 2*B*b*d)/b**5 -
(a*e - b*d)**(5/2)*(7*A*b*e - 9*B*a*e + 2*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqr
t(a*e - b*d))/b**(11/2)

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Mathematica [A]  time = 1.31853, size = 252, normalized size = 0.98 \[ \frac{\sqrt{d+e x} \left (-840 a^3 B e^3+2 b e x \left (105 a^2 B e^2-14 a b e (5 A e+16 B d)+2 b^2 d (56 A e+61 B d)\right )+210 a^2 b e^2 (3 A e+10 B d)+6 b^2 e^2 x^2 (-14 a B e+7 A b e+22 b B d)-56 a b^2 d e (25 A e+29 B d)-\frac{105 (A b-a B) (b d-a e)^3}{a+b x}+4 b^3 d^2 (203 A e+88 B d)+30 b^3 B e^3 x^3\right )}{105 b^5}-\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(Sqrt[d + e*x]*(-840*a^3*B*e^3 + 210*a^2*b*e^2*(10*B*d + 3*A*e) - 56*a*b^2*d*e*(
29*B*d + 25*A*e) + 4*b^3*d^2*(88*B*d + 203*A*e) + 2*b*e*(105*a^2*B*e^2 - 14*a*b*
e*(16*B*d + 5*A*e) + 2*b^2*d*(61*B*d + 56*A*e))*x + 6*b^2*e^2*(22*b*B*d + 7*A*b*
e - 14*a*B*e)*x^2 + 30*b^3*B*e^3*x^3 - (105*(A*b - a*B)*(b*d - a*e)^3)/(a + b*x)
))/(105*b^5) - ((b*d - a*e)^(5/2)*(2*b*B*d + 7*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]
*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

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Maple [B]  time = 0.045, size = 915, normalized size = 3.6 \[ \text{result too large to display} \]

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),x)

[Out]

3/b^4*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a^3*d*e^3-3/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*B*d^
2*a^2*e^2+1/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a*d^3*e+21/b^3/(b*(a*e-b*d))^(1/2)*a
rctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A*d*a^2*e^3+2/5/b^2*A*(e*x+d)^(5/2)*e
+2/5/b^2*B*(e*x+d)^(5/2)*d+2/3/b^2*B*(e*x+d)^(3/2)*d^2+2/b^2*B*d^3*(e*x+d)^(1/2)
-8/3/b^3*B*(e*x+d)^(3/2)*a*d*e-12/b^3*A*a*d*e^2*(e*x+d)^(1/2)+18/b^4*B*a^2*d*e^2
*(e*x+d)^(1/2)-12/b^3*B*a*d^2*e*(e*x+d)^(1/2)+1/b^4*(e*x+d)^(1/2)/(b*e*x+a*e)*A*
a^3*e^4-1/b*(e*x+d)^(1/2)/(b*e*x+a*e)*A*d^3*e-1/b^5*(e*x+d)^(1/2)/(b*e*x+a*e)*B*
e^4*a^4-7/b^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A*
a^3*e^4+7/b/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A*d^
3*e+9/b^5/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*e^4*
a^4+33/b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*d^2
*a^2*e^2-15/b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*
B*a*d^3*e-21/b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*A*a*d^2*e^2-29/b^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/
2))*B*a^3*d*e^3-3/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*A*d*a^2*e^3+3/b^2*(e*x+d)^(1/2)/
(b*e*x+a*e)*A*a*d^2*e^2+2/7/b^2*B*(e*x+d)^(7/2)+4/3/b^2*A*(e*x+d)^(3/2)*d*e+2/b/
(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*d^4+6/b^2*A*d^
2*e*(e*x+d)^(1/2)-8/b^5*B*e^3*a^3*(e*x+d)^(1/2)+2/b^4*B*(e*x+d)^(3/2)*a^2*e^2+6/
b^4*A*a^2*e^3*(e*x+d)^(1/2)-4/5/b^3*B*(e*x+d)^(5/2)*a*e-4/3/b^3*A*(e*x+d)^(3/2)*
a*e^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)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.307017, 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),x, algorithm="fricas")

[Out]

[1/210*(105*(2*B*a*b^3*d^3 - (13*B*a^2*b^2 - 7*A*a*b^3)*d^2*e + 2*(10*B*a^3*b -
7*A*a^2*b^2)*d*e^2 - (9*B*a^4 - 7*A*a^3*b)*e^3 + (2*B*b^4*d^3 - (13*B*a*b^3 - 7*
A*b^4)*d^2*e + 2*(10*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 - (9*B*a^3*b - 7*A*a^2*b^2)*e^
3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d
- a*e)/b))/(b*x + a)) + 2*(30*B*b^4*e^3*x^4 + (457*B*a*b^3 - 105*A*b^4)*d^3 - 7
*(277*B*a^2*b^2 - 161*A*a*b^3)*d^2*e + 35*(69*B*a^3*b - 49*A*a^2*b^2)*d*e^2 - 10
5*(9*B*a^4 - 7*A*a^3*b)*e^3 + 6*(22*B*b^4*d*e^2 - (9*B*a*b^3 - 7*A*b^4)*e^3)*x^3
+ 2*(122*B*b^4*d^2*e - 2*(79*B*a*b^3 - 56*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 7*A*a
*b^3)*e^3)*x^2 + 2*(176*B*b^4*d^3 - 2*(345*B*a*b^3 - 203*A*b^4)*d^2*e + 14*(59*B
*a^2*b^2 - 42*A*a*b^3)*d*e^2 - 35*(9*B*a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d
))/(b^6*x + a*b^5), -1/105*(105*(2*B*a*b^3*d^3 - (13*B*a^2*b^2 - 7*A*a*b^3)*d^2*
e + 2*(10*B*a^3*b - 7*A*a^2*b^2)*d*e^2 - (9*B*a^4 - 7*A*a^3*b)*e^3 + (2*B*b^4*d^
3 - (13*B*a*b^3 - 7*A*b^4)*d^2*e + 2*(10*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 - (9*B*a^3
*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d -
a*e)/b)) - (30*B*b^4*e^3*x^4 + (457*B*a*b^3 - 105*A*b^4)*d^3 - 7*(277*B*a^2*b^2
- 161*A*a*b^3)*d^2*e + 35*(69*B*a^3*b - 49*A*a^2*b^2)*d*e^2 - 105*(9*B*a^4 - 7*
A*a^3*b)*e^3 + 6*(22*B*b^4*d*e^2 - (9*B*a*b^3 - 7*A*b^4)*e^3)*x^3 + 2*(122*B*b^4
*d^2*e - 2*(79*B*a*b^3 - 56*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 7*A*a*b^3)*e^3)*x^2
+ 2*(176*B*b^4*d^3 - 2*(345*B*a*b^3 - 203*A*b^4)*d^2*e + 14*(59*B*a^2*b^2 - 42*A
*a*b^3)*d*e^2 - 35*(9*B*a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^6*x + a*b
^5)]

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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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.332465, size = 815, normalized size = 3.18 \[ \frac{{\left (2 \, B b^{4} d^{4} - 15 \, B a b^{3} d^{3} e + 7 \, A b^{4} d^{3} e + 33 \, B a^{2} b^{2} d^{2} e^{2} - 21 \, A a b^{3} d^{2} e^{2} - 29 \, B a^{3} b d e^{3} + 21 \, A a^{2} b^{2} d e^{3} + 9 \, B a^{4} e^{4} - 7 \, A a^{3} b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{5}} + \frac{\sqrt{x e + d} B a b^{3} d^{3} e - \sqrt{x e + d} A b^{4} d^{3} e - 3 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{2} + 3 \, \sqrt{x e + d} A a b^{3} d^{2} e^{2} + 3 \, \sqrt{x e + d} B a^{3} b d e^{3} - 3 \, \sqrt{x e + d} A a^{2} b^{2} d e^{3} - \sqrt{x e + d} B a^{4} e^{4} + \sqrt{x e + d} A a^{3} b e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{12} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{12} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{12} d^{2} + 105 \, \sqrt{x e + d} B b^{12} d^{3} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{11} e + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{12} e - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{11} d e + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{12} d e - 630 \, \sqrt{x e + d} B a b^{11} d^{2} e + 315 \, \sqrt{x e + d} A b^{12} d^{2} e + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{10} e^{2} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{11} e^{2} + 945 \, \sqrt{x e + d} B a^{2} b^{10} d e^{2} - 630 \, \sqrt{x e + d} A a b^{11} d e^{2} - 420 \, \sqrt{x e + d} B a^{3} b^{9} e^{3} + 315 \, \sqrt{x e + d} A a^{2} b^{10} e^{3}\right )}}{105 \, b^{14}} \]

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),x, algorithm="giac")

[Out]

(2*B*b^4*d^4 - 15*B*a*b^3*d^3*e + 7*A*b^4*d^3*e + 33*B*a^2*b^2*d^2*e^2 - 21*A*a*
b^3*d^2*e^2 - 29*B*a^3*b*d*e^3 + 21*A*a^2*b^2*d*e^3 + 9*B*a^4*e^4 - 7*A*a^3*b*e^
4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) + (sq
rt(x*e + d)*B*a*b^3*d^3*e - sqrt(x*e + d)*A*b^4*d^3*e - 3*sqrt(x*e + d)*B*a^2*b^
2*d^2*e^2 + 3*sqrt(x*e + d)*A*a*b^3*d^2*e^2 + 3*sqrt(x*e + d)*B*a^3*b*d*e^3 - 3*
sqrt(x*e + d)*A*a^2*b^2*d*e^3 - sqrt(x*e + d)*B*a^4*e^4 + sqrt(x*e + d)*A*a^3*b*
e^4)/(((x*e + d)*b - b*d + a*e)*b^5) + 2/105*(15*(x*e + d)^(7/2)*B*b^12 + 21*(x*
e + d)^(5/2)*B*b^12*d + 35*(x*e + d)^(3/2)*B*b^12*d^2 + 105*sqrt(x*e + d)*B*b^12
*d^3 - 42*(x*e + d)^(5/2)*B*a*b^11*e + 21*(x*e + d)^(5/2)*A*b^12*e - 140*(x*e +
d)^(3/2)*B*a*b^11*d*e + 70*(x*e + d)^(3/2)*A*b^12*d*e - 630*sqrt(x*e + d)*B*a*b^
11*d^2*e + 315*sqrt(x*e + d)*A*b^12*d^2*e + 105*(x*e + d)^(3/2)*B*a^2*b^10*e^2 -
70*(x*e + d)^(3/2)*A*a*b^11*e^2 + 945*sqrt(x*e + d)*B*a^2*b^10*d*e^2 - 630*sqrt
(x*e + d)*A*a*b^11*d*e^2 - 420*sqrt(x*e + d)*B*a^3*b^9*e^3 + 315*sqrt(x*e + d)*A
*a^2*b^10*e^3)/b^14