3.1741 \(\int \frac{(A+B x) (d+e x)^{7/2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=274 \[ -\frac{7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2}}+\frac{7 e \sqrt{d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5}+\frac{7 e (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4}+\frac{7 e (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/(4*b^5) + (7*e*(4*
b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(3/2))/(12*b^4) + (7*e*(4*b*B*d + 5*A*b*e -
9*a*B*e)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)) - ((4*b*B*d + 5*A*b*e - 9*a*B*e)
*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(9/2))/
(2*b*(b*d - a*e)*(a + b*x)^2) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d + 5*A*b*e - 9*a*
B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(11/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.593197, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2}}+\frac{7 e \sqrt{d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5}+\frac{7 e (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4}+\frac{7 e (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^3,x]

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/(4*b^5) + (7*e*(4*
b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(3/2))/(12*b^4) + (7*e*(4*b*B*d + 5*A*b*e -
9*a*B*e)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)) - ((4*b*B*d + 5*A*b*e - 9*a*B*e)
*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(9/2))/
(2*b*(b*d - a*e)*(a + b*x)^2) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d + 5*A*b*e - 9*a*
B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(11/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 60.7287, size = 269, normalized size = 0.98 \[ \frac{\left (d + e x\right )^{\frac{9}{2}} \left (A b - B a\right )}{2 b \left (a + b x\right )^{2} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (5 A b e - 9 B a e + 4 B b d\right )}{4 b^{2} \left (a + b x\right ) \left (a e - b d\right )} - \frac{7 e \left (d + e x\right )^{\frac{5}{2}} \left (5 A b e - 9 B a e + 4 B b d\right )}{20 b^{3} \left (a e - b d\right )} + \frac{7 e \left (d + e x\right )^{\frac{3}{2}} \left (5 A b e - 9 B a e + 4 B b d\right )}{12 b^{4}} - \frac{7 e \sqrt{d + e x} \left (a e - b d\right ) \left (5 A b e - 9 B a e + 4 B b d\right )}{4 b^{5}} + \frac{7 e \left (a e - b d\right )^{\frac{3}{2}} \left (5 A b e - 9 B a e + 4 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a)**3,x)

[Out]

(d + e*x)**(9/2)*(A*b - B*a)/(2*b*(a + b*x)**2*(a*e - b*d)) + (d + e*x)**(7/2)*(
5*A*b*e - 9*B*a*e + 4*B*b*d)/(4*b**2*(a + b*x)*(a*e - b*d)) - 7*e*(d + e*x)**(5/
2)*(5*A*b*e - 9*B*a*e + 4*B*b*d)/(20*b**3*(a*e - b*d)) + 7*e*(d + e*x)**(3/2)*(5
*A*b*e - 9*B*a*e + 4*B*b*d)/(12*b**4) - 7*e*sqrt(d + e*x)*(a*e - b*d)*(5*A*b*e -
9*B*a*e + 4*B*b*d)/(4*b**5) + 7*e*(a*e - b*d)**(3/2)*(5*A*b*e - 9*B*a*e + 4*B*b
*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(4*b**(11/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.805434, size = 223, normalized size = 0.81 \[ \frac{\sqrt{d+e x} \left (8 e \left (90 a^2 B e^2-15 a b e (3 A e+10 B d)+2 b^2 d (25 A e+29 B d)\right )+8 b e^2 x (-15 a B e+5 A b e+16 b B d)-\frac{15 (b d-a e)^2 (-17 a B e+13 A b e+4 b B d)}{a+b x}-\frac{30 (A b-a B) (b d-a e)^3}{(a+b x)^2}+24 b^2 B e^3 x^2\right )}{60 b^5}-\frac{7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^3,x]

[Out]

(Sqrt[d + e*x]*(8*e*(90*a^2*B*e^2 - 15*a*b*e*(10*B*d + 3*A*e) + 2*b^2*d*(29*B*d
+ 25*A*e)) + 8*b*e^2*(16*b*B*d + 5*A*b*e - 15*a*B*e)*x + 24*b^2*B*e^3*x^2 - (30*
(A*b - a*B)*(b*d - a*e)^3)/(a + b*x)^2 - (15*(b*d - a*e)^2*(4*b*B*d + 13*A*b*e -
17*a*B*e))/(a + b*x)))/(60*b^5) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d + 5*A*b*e - 9
*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(11/2))

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 940, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^3,x)

[Out]

-19/2/b^3/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a^2*d*e^3-13/4/b/(b*e*x+a*e)^2*(e*x+d)^(
3/2)*A*d^2*e^2+17/4/b^4/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a^3*e^4-18/b^4*B*a*d*e^2*(
e*x+d)^(1/2)-13/4/b^3/(b*e*x+a*e)^2*(e*x+d)^(3/2)*A*a^2*e^4-11/4/b^4/(b*e*x+a*e)
^2*(e*x+d)^(1/2)*A*a^3*e^5+11/4/b/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*d^3*e^2+15/4/b^5
/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^4*e^5+35/4/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+
d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*a^2*e^4+35/4/b^2/((a*e-b*d)*b)^(1/2)*arctan((e
*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*d^2*e^2-63/4/b^5/((a*e-b*d)*b)^(1/2)*arctan
((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a^3*e^4+2/3/b^3*A*(e*x+d)^(3/2)*e^2+2/5*
e/b^3*B*(e*x+d)^(5/2)+6*e/b^3*B*d^2*(e*x+d)^(1/2)-27/4/b^2/(b*e*x+a*e)^2*(e*x+d)
^(1/2)*B*a*d^3*e^2-2/b^4*B*(e*x+d)^(3/2)*a*e^2-6/b^4*A*a*e^3*(e*x+d)^(1/2)+6/b^3
*A*d*e^2*(e*x+d)^(1/2)+12/b^5*B*a^2*e^3*(e*x+d)^(1/2)+4/3*e/b^3*B*(e*x+d)^(3/2)*
d+25/4/b^2/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a*d^2*e^2-35/2/b^3/((a*e-b*d)*b)^(1/2)*
arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*a*d*e^3+77/2/b^4/((a*e-b*d)*b)^(1/
2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a^2*d*e^3-119/4/b^3/((a*e-b*d)*
b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a*d^2*e^2+13/2/b^2/(b*e*x
+a*e)^2*(e*x+d)^(3/2)*A*a*d*e^3-e/b/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*d^3+e/b/(b*e*x
+a*e)^2*(e*x+d)^(1/2)*B*d^4+7*e/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/(
(a*e-b*d)*b)^(1/2))*B*d^3+57/4/b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^2*d^2*e^3+33/
4/b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a^2*d*e^4-33/4/b^2/(b*e*x+a*e)^2*(e*x+d)^(1/
2)*A*a*d^2*e^3-49/4/b^4/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^3*d*e^4

_______________________________________________________________________________________

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*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.232953, 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*x + a)^3,x, algorithm="fricas")

[Out]

[1/120*(105*(4*B*a^2*b^2*d^2*e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5
*A*a^3*b)*e^3 + (4*B*b^4*d^2*e - (13*B*a*b^3 - 5*A*b^4)*d*e^2 + (9*B*a^2*b^2 - 5
*A*a*b^3)*e^3)*x^2 + 2*(4*B*a*b^3*d^2*e - (13*B*a^2*b^2 - 5*A*a*b^3)*d*e^2 + (9*
B*a^3*b - 5*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*(24*B*b^4*e^3*x^4 - 30*(B*a*
b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15*A*a*b^3)*d^2*e - 140*(12*B*a^3*b - 5*A*
a^2*b^2)*d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e^2 - (9*B*a*b^3
- 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e - 2*(59*B*a*b^3 - 25*A*b^4)*d*e^2 + 7*(9
*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3 - 195*A*b^4)*d^
2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*(9*B*a^3*b - 5*A*a^2*b^2)*e^3)
*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), -1/60*(105*(4*B*a^2*b^2*d^2*
e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5*A*a^3*b)*e^3 + (4*B*b^4*d^2*
e - (13*B*a*b^3 - 5*A*b^4)*d*e^2 + (9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + 2*(4*B*a
*b^3*d^2*e - (13*B*a^2*b^2 - 5*A*a*b^3)*d*e^2 + (9*B*a^3*b - 5*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)) - (24*B*b^4*e^
3*x^4 - 30*(B*a*b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15*A*a*b^3)*d^2*e - 140*(1
2*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e
^2 - (9*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e - 2*(59*B*a*b^3 - 25*A*b
^4)*d*e^2 + 7*(9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3
- 195*A*b^4)*d^2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*(9*B*a^3*b - 5
*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)]

_______________________________________________________________________________________

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*x+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234489, size = 819, normalized size = 2.99 \[ \frac{7 \,{\left (4 \, B b^{3} d^{3} e - 17 \, B a b^{2} d^{2} e^{2} + 5 \, A b^{3} d^{2} e^{2} + 22 \, B a^{2} b d e^{3} - 10 \, A a b^{2} d e^{3} - 9 \, B a^{3} e^{4} + 5 \, A a^{2} b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{5}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{3} e - 4 \, \sqrt{x e + d} B b^{4} d^{4} e - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d^{2} e^{2} + 13 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d^{2} e^{2} + 27 \, \sqrt{x e + d} B a b^{3} d^{3} e^{2} - 11 \, \sqrt{x e + d} A b^{4} d^{3} e^{2} + 38 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} d e^{3} - 26 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} d e^{3} - 57 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{3} + 33 \, \sqrt{x e + d} A a b^{3} d^{2} e^{3} - 17 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b e^{4} + 13 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{2} e^{4} + 49 \, \sqrt{x e + d} B a^{3} b d e^{4} - 33 \, \sqrt{x e + d} A a^{2} b^{2} d e^{4} - 15 \, \sqrt{x e + d} B a^{4} e^{5} + 11 \, \sqrt{x e + d} A a^{3} b e^{5}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{12} e + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{12} d e + 45 \, \sqrt{x e + d} B b^{12} d^{2} e - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{11} e^{2} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{12} e^{2} - 135 \, \sqrt{x e + d} B a b^{11} d e^{2} + 45 \, \sqrt{x e + d} A b^{12} d e^{2} + 90 \, \sqrt{x e + d} B a^{2} b^{10} e^{3} - 45 \, \sqrt{x e + d} A a b^{11} e^{3}\right )}}{15 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(b*x + a)^3,x, algorithm="giac")

[Out]

7/4*(4*B*b^3*d^3*e - 17*B*a*b^2*d^2*e^2 + 5*A*b^3*d^2*e^2 + 22*B*a^2*b*d*e^3 - 1
0*A*a*b^2*d*e^3 - 9*B*a^3*e^4 + 5*A*a^2*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) - 1/4*(4*(x*e + d)^(3/2)*B*b^4*d^3*e - 4*
sqrt(x*e + d)*B*b^4*d^4*e - 25*(x*e + d)^(3/2)*B*a*b^3*d^2*e^2 + 13*(x*e + d)^(3
/2)*A*b^4*d^2*e^2 + 27*sqrt(x*e + d)*B*a*b^3*d^3*e^2 - 11*sqrt(x*e + d)*A*b^4*d^
3*e^2 + 38*(x*e + d)^(3/2)*B*a^2*b^2*d*e^3 - 26*(x*e + d)^(3/2)*A*a*b^3*d*e^3 -
57*sqrt(x*e + d)*B*a^2*b^2*d^2*e^3 + 33*sqrt(x*e + d)*A*a*b^3*d^2*e^3 - 17*(x*e
+ d)^(3/2)*B*a^3*b*e^4 + 13*(x*e + d)^(3/2)*A*a^2*b^2*e^4 + 49*sqrt(x*e + d)*B*a
^3*b*d*e^4 - 33*sqrt(x*e + d)*A*a^2*b^2*d*e^4 - 15*sqrt(x*e + d)*B*a^4*e^5 + 11*
sqrt(x*e + d)*A*a^3*b*e^5)/(((x*e + d)*b - b*d + a*e)^2*b^5) + 2/15*(3*(x*e + d)
^(5/2)*B*b^12*e + 10*(x*e + d)^(3/2)*B*b^12*d*e + 45*sqrt(x*e + d)*B*b^12*d^2*e
- 15*(x*e + d)^(3/2)*B*a*b^11*e^2 + 5*(x*e + d)^(3/2)*A*b^12*e^2 - 135*sqrt(x*e
+ d)*B*a*b^11*d*e^2 + 45*sqrt(x*e + d)*A*b^12*d*e^2 + 90*sqrt(x*e + d)*B*a^2*b^1
0*e^3 - 45*sqrt(x*e + d)*A*a*b^11*e^3)/b^15