3.1229 \(\int \frac{(A+B x) (d+e x)^{7/2}}{b x+c x^2} \, dx\)

Optimal. Leaf size=228 \[ \frac{2 \sqrt{d+e x} \left (A c e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )+B (c d-b e)^3\right )}{c^4}-\frac{2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{9/2}}+\frac{2 (d+e x)^{3/2} \left (A c e (2 c d-b e)+B (c d-b e)^2\right )}{3 c^3}+\frac{2 (d+e x)^{5/2} (A c e-b B e+B c d)}{5 c^2}-\frac{2 A d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B (d+e x)^{7/2}}{7 c} \]

[Out]

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

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Rubi [A]  time = 1.29533, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 \sqrt{d+e x} \left (A c e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )+B (c d-b e)^3\right )}{c^4}-\frac{2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{9/2}}+\frac{2 (d+e x)^{3/2} \left (A c e (2 c d-b e)+B (c d-b e)^2\right )}{3 c^3}+\frac{2 (d+e x)^{5/2} (A c e-b B e+B c d)}{5 c^2}-\frac{2 A d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B (d+e x)^{7/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 129.19, size = 224, normalized size = 0.98 \[ - \frac{2 A d^{\frac{7}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b} + \frac{2 B \left (d + e x\right )^{\frac{7}{2}}}{7 c} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A c e - B b e + B c d\right )}{5 c^{2}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A c^{2} d e + \left (b e - c d\right ) \left (- A c e + B \left (b e - c d\right )\right )\right )}{3 c^{3}} - \frac{2 \sqrt{d + e x} \left (- A c^{3} d^{2} e + \left (b e - c d\right ) \left (A c^{2} d e + \left (b e - c d\right ) \left (- A c e + B \left (b e - c d\right )\right )\right )\right )}{c^{4}} - \frac{2 \left (A c - B b\right ) \left (b e - c d\right )^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(7/2)/(c*x**2+b*x),x)

[Out]

-2*A*d**(7/2)*atanh(sqrt(d + e*x)/sqrt(d))/b + 2*B*(d + e*x)**(7/2)/(7*c) + 2*(d
+ e*x)**(5/2)*(A*c*e - B*b*e + B*c*d)/(5*c**2) + 2*(d + e*x)**(3/2)*(A*c**2*d*e
+ (b*e - c*d)*(-A*c*e + B*(b*e - c*d)))/(3*c**3) - 2*sqrt(d + e*x)*(-A*c**3*d**
2*e + (b*e - c*d)*(A*c**2*d*e + (b*e - c*d)*(-A*c*e + B*(b*e - c*d))))/c**4 - 2*
(A*c - B*b)*(b*e - c*d)**(7/2)*atan(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b*c*
*(9/2))

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Mathematica [A]  time = 0.438989, size = 239, normalized size = 1.05 \[ \frac{2 \sqrt{d+e x} \left (7 A c e \left (15 b^2 e^2-5 b c e (10 d+e x)+c^2 \left (58 d^2+16 d e x+3 e^2 x^2\right )\right )+B \left (-105 b^3 e^3+35 b^2 c e^2 (10 d+e x)-7 b c^2 e \left (58 d^2+16 d e x+3 e^2 x^2\right )+c^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{105 c^4}+\frac{2 (A c-b B) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{9/2}}-\frac{2 A d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(7*A*c*e*(15*b^2*e^2 - 5*b*c*e*(10*d + e*x) + c^2*(58*d^2 + 16*
d*e*x + 3*e^2*x^2)) + B*(-105*b^3*e^3 + 35*b^2*c*e^2*(10*d + e*x) - 7*b*c^2*e*(5
8*d^2 + 16*d*e*x + 3*e^2*x^2) + c^3*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 + 15*e
^3*x^3))))/(105*c^4) - (2*A*d^(7/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b + (2*(-(b*
B) + A*c)*(c*d - b*e)^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b
*c^(9/2))

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Maple [B]  time = 0.028, size = 741, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b+2/7*B*(e*x+d)^(7/2)/c+8*b^2/c^2/((
b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d*e^3-12*b/c/((b
*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^2*e^2-8*b^3/c^3
/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d*e^3+12*b^2/
c^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^2*e^2-8*
b/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^3*e-2/c^
4*B*e^3*b^3*(e*x+d)^(1/2)+8/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d
)*c)^(1/2))*A*d^3*e-4/3/c^2*B*(e*x+d)^(3/2)*b*d*e-6/c^2*A*b*d*e^2*(e*x+d)^(1/2)+
6/c^3*B*b^2*d*e^2*(e*x+d)^(1/2)-6/c^2*B*b*d^2*e*(e*x+d)^(1/2)-2*b^3/c^3/((b*e-c*
d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*e^4-2/b*c/((b*e-c*d)*c
)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^4+2*b^4/c^4/((b*e-c*d)*c
)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*e^4+2/5/c*A*(e*x+d)^(5/2)*
e+2/5/c*B*(e*x+d)^(5/2)*d+2/3/c*B*(e*x+d)^(3/2)*d^2+2/c*B*d^3*(e*x+d)^(1/2)+2/((
b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^4-2/5/c^2*B*(e
*x+d)^(5/2)*b*e-2/3/c^2*A*(e*x+d)^(3/2)*b*e^2+4/3/c*A*(e*x+d)^(3/2)*d*e+2/3/c^3*
B*(e*x+d)^(3/2)*b^2*e^2+2/c^3*A*b^2*e^3*(e*x+d)^(1/2)+6/c*A*d^2*e*(e*x+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)*(e*x + d)^(7/2)/(c*x^2 + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.5629, 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)/(c*x^2 + b*x),x, algorithm="fricas")

[Out]

[1/105*(105*A*c^4*d^(7/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 105*((B
*b*c^3 - A*c^4)*d^3 - 3*(B*b^2*c^2 - A*b*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)*d*
e^2 - (B*b^4 - A*b^3*c)*e^3)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sq
rt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 2*(15*B*b*c^3*e^3*x^3 + 176*B*b*
c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^3*c - A*b^2*c^2)*d*e^2 - 10
5*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*b^2*c^2 - A*b*c^3)*e^3)*x^2
+ (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e^2 + 35*(B*b^3*c - A*b^2*c^
2)*e^3)*x)*sqrt(e*x + d))/(b*c^4), 1/105*(105*A*c^4*d^(7/2)*log((e*x - 2*sqrt(e*
x + d)*sqrt(d) + 2*d)/x) - 210*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2*c^2 - A*b*c^3)*
d^2*e + 3*(B*b^3*c - A*b^2*c^2)*d*e^2 - (B*b^4 - A*b^3*c)*e^3)*sqrt(-(c*d - b*e)
/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + 2*(15*B*b*c^3*e^3*x^3 + 176*B*b
*c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^3*c - A*b^2*c^2)*d*e^2 - 1
05*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*b^2*c^2 - A*b*c^3)*e^3)*x^
2 + (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e^2 + 35*(B*b^3*c - A*b^2*c
^2)*e^3)*x)*sqrt(e*x + d))/(b*c^4), -1/105*(210*A*c^4*sqrt(-d)*d^3*arctan(sqrt(e
*x + d)/sqrt(-d)) + 105*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2*c^2 - A*b*c^3)*d^2*e +
3*(B*b^3*c - A*b^2*c^2)*d*e^2 - (B*b^4 - A*b^3*c)*e^3)*sqrt((c*d - b*e)/c)*log(
(c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 2*(15
*B*b*c^3*e^3*x^3 + 176*B*b*c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^
3*c - A*b^2*c^2)*d*e^2 - 105*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*
b^2*c^2 - A*b*c^3)*e^3)*x^2 + (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e
^2 + 35*(B*b^3*c - A*b^2*c^2)*e^3)*x)*sqrt(e*x + d))/(b*c^4), -2/105*(105*A*c^4*
sqrt(-d)*d^3*arctan(sqrt(e*x + d)/sqrt(-d)) + 105*((B*b*c^3 - A*c^4)*d^3 - 3*(B*
b^2*c^2 - A*b*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)*d*e^2 - (B*b^4 - A*b^3*c)*e^3
)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) - (15*B*b*c^3*
e^3*x^3 + 176*B*b*c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^3*c - A*b
^2*c^2)*d*e^2 - 105*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*b^2*c^2 -
A*b*c^3)*e^3)*x^2 + (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e^2 + 35*(
B*b^3*c - A*b^2*c^2)*e^3)*x)*sqrt(e*x + d))/(b*c^4)]

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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)/(c*x**2+b*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.294862, size = 643, normalized size = 2.82 \[ \frac{2 \, A d^{4} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} + \frac{2 \,{\left (B b c^{4} d^{4} - A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - A b^{4} c e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b c^{4}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{6} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{6} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{6} d^{2} + 105 \, \sqrt{x e + d} B c^{6} d^{3} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B b c^{5} e + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{6} e - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} B b c^{5} d e + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{6} d e - 315 \, \sqrt{x e + d} B b c^{5} d^{2} e + 315 \, \sqrt{x e + d} A c^{6} d^{2} e + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} c^{4} e^{2} - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A b c^{5} e^{2} + 315 \, \sqrt{x e + d} B b^{2} c^{4} d e^{2} - 315 \, \sqrt{x e + d} A b c^{5} d e^{2} - 105 \, \sqrt{x e + d} B b^{3} c^{3} e^{3} + 105 \, \sqrt{x e + d} A b^{2} c^{4} e^{3}\right )}}{105 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x),x, algorithm="giac")

[Out]

2*A*d^4*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)) + 2*(B*b*c^4*d^4 - A*c^5*d^4
- 4*B*b^2*c^3*d^3*e + 4*A*b*c^4*d^3*e + 6*B*b^3*c^2*d^2*e^2 - 6*A*b^2*c^3*d^2*e
^2 - 4*B*b^4*c*d*e^3 + 4*A*b^3*c^2*d*e^3 + B*b^5*e^4 - A*b^4*c*e^4)*arctan(sqrt(
x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b*c^4) + 2/105*(15*(x*e +
d)^(7/2)*B*c^6 + 21*(x*e + d)^(5/2)*B*c^6*d + 35*(x*e + d)^(3/2)*B*c^6*d^2 + 10
5*sqrt(x*e + d)*B*c^6*d^3 - 21*(x*e + d)^(5/2)*B*b*c^5*e + 21*(x*e + d)^(5/2)*A*
c^6*e - 70*(x*e + d)^(3/2)*B*b*c^5*d*e + 70*(x*e + d)^(3/2)*A*c^6*d*e - 315*sqrt
(x*e + d)*B*b*c^5*d^2*e + 315*sqrt(x*e + d)*A*c^6*d^2*e + 35*(x*e + d)^(3/2)*B*b
^2*c^4*e^2 - 35*(x*e + d)^(3/2)*A*b*c^5*e^2 + 315*sqrt(x*e + d)*B*b^2*c^4*d*e^2
- 315*sqrt(x*e + d)*A*b*c^5*d*e^2 - 105*sqrt(x*e + d)*B*b^3*c^3*e^3 + 105*sqrt(x
*e + d)*A*b^2*c^4*e^3)/c^7