3.1873 \(\int \frac{A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=414 \[ -\frac{7 b e (a+b x) (5 a B e-9 A b e+4 b B d)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}-\frac{7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{12 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac{5 a B e-9 A b e+4 b B d}{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{20 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac{A b-a B}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac{7 b^{3/2} e (a+b x) (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \]

[Out]

-(4*b*B*d - 9*A*b*e + 5*a*B*e)/(4*b*(b*d - a*e)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2]) - (A*b - a*B)/(2*b*(b*d - a*e)*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - (7*e*(4*b*B*d - 9*A*b*e + 5*a*B*e)*(a + b*x))/(20*b*(
b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(4*b*B*d - 9*
A*b*e + 5*a*B*e)*(a + b*x))/(12*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - (7*b*e*(4*b*B*d - 9*A*b*e + 5*a*B*e)*(a + b*x))/(4*(b*d - a*e)^5*
Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*b^(3/2)*e*(4*b*B*d - 9*A*b*e +
5*a*B*e)*(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 = 0.934305, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{7 b e (a+b x) (5 a B e-9 A b e+4 b B d)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}-\frac{7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{12 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac{5 a B e-9 A b e+4 b B d}{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{20 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac{A b-a B}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac{7 b^{3/2} e (a+b x) (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(4*b*B*d - 9*A*b*e + 5*a*B*e)/(4*b*(b*d - a*e)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2]) - (A*b - a*B)/(2*b*(b*d - a*e)*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - (7*e*(4*b*B*d - 9*A*b*e + 5*a*B*e)*(a + b*x))/(20*b*(
b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(4*b*B*d - 9*
A*b*e + 5*a*B*e)*(a + b*x))/(12*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - (7*b*e*(4*b*B*d - 9*A*b*e + 5*a*B*e)*(a + b*x))/(4*(b*d - a*e)^5*
Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*b^(3/2)*e*(4*b*B*d - 9*A*b*e +
5*a*B*e)*(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((B*x+A)/(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 = 2.0237, size = 256, normalized size = 0.62 \[ \frac{(a+b x)^3 \left (\frac{7 b^{3/2} e (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{11/2}}+\frac{\sqrt{d+e x} \left (-\frac{15 b^2 (11 a B e-15 A b e+4 b B d)}{a+b x}-\frac{30 b^2 (A b-a B) (b d-a e)}{(a+b x)^2}+\frac{360 b e (-a B e+2 A b e-b B d)}{d+e x}+\frac{40 e (b d-a e) (-a B e+3 A b e-2 b B d)}{(d+e x)^2}+\frac{24 e (b d-a e)^2 (A e-B d)}{(d+e x)^3}\right )}{15 (b d-a e)^5}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^3*((Sqrt[d + e*x]*((-30*b^2*(A*b - a*B)*(b*d - a*e))/(a + b*x)^2 - (1
5*b^2*(4*b*B*d - 15*A*b*e + 11*a*B*e))/(a + b*x) + (24*e*(b*d - a*e)^2*(-(B*d) +
A*e))/(d + e*x)^3 + (40*e*(b*d - a*e)*(-2*b*B*d + 3*A*b*e - a*B*e))/(d + e*x)^2
+ (360*b*e*(-(b*B*d) + 2*A*b*e - a*B*e))/(d + e*x)))/(15*(b*d - a*e)^5) + (7*b^
(3/2)*e*(4*b*B*d - 9*A*b*e + 5*a*B*e)*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 = 0.045, size = 1230, normalized size = 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)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

-1/60*(945*A*(b*(a*e-b*d))^(1/2)*x^4*b^4*e^4+40*B*(b*(a*e-b*d))^(1/2)*x*a^4*e^4-
60*B*(b*(a*e-b*d))^(1/2)*x*b^4*d^4-72*A*(b*(a*e-b*d))^(1/2)*x*a^3*b*e^4+135*A*(b
*(a*e-b*d))^(1/2)*x*b^4*d^3*e+945*A*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+24*A*(b*(a*e-b*d))^(1/2)*a^4*e^4-30*A*(b*(a*e-b*d))^(1
/2)*b^4*d^4+3717*A*(b*(a*e-b*d))^(1/2)*x^2*a*b^3*d*e^3-2289*B*(b*(a*e-b*d))^(1/2
)*x^2*a^2*b^2*d*e^3-2457*B*(b*(a*e-b*d))^(1/2)*x^2*a*b^3*d^2*e^2+1224*A*(b*(a*e-
b*d))^(1/2)*x*a^2*b^2*d*e^3+2493*A*(b*(a*e-b*d))^(1/2)*x*a*b^3*d^2*e^2-648*B*(b*
(a*e-b*d))^(1/2)*x*a^3*b*d*e^3-1929*B*(b*(a*e-b*d))^(1/2)*x*a^2*b^2*d^2*e^2-1183
*B*(b*(a*e-b*d))^(1/2)*x*a*b^3*d^3*e-525*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^
(1/2))*(e*x+d)^(5/2)*x^2*a*b^4*e^2-420*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1
/2))*(e*x+d)^(5/2)*x^2*b^5*d*e+1890*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2)
)*(e*x+d)^(5/2)*x*a*b^4*e^2-1050*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(
e*x+d)^(5/2)*x*a^2*b^3*e^2-420*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*
x+d)^(5/2)*a^2*b^3*d*e-1925*B*(b*(a*e-b*d))^(1/2)*x^3*a*b^3*d*e^3-840*B*arctan((
e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*x*a*b^4*d*e+945*A*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*(b*(a*e-b*d))^(1
/2)*x^4*a*b^3*e^4-420*B*(b*(a*e-b*d))^(1/2)*x^4*b^4*d*e^3-525*B*arctan((e*x+d)^(
1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*a^3*b^2*e^2+1575*A*(b*(a*e-b*d))^(1/2)
*x^3*a*b^3*e^4+2205*A*(b*(a*e-b*d))^(1/2)*x^3*b^4*d*e^3-875*B*(b*(a*e-b*d))^(1/2
)*x^3*a^2*b^2*e^4-980*B*(b*(a*e-b*d))^(1/2)*x^3*b^4*d^2*e^2+504*A*(b*(a*e-b*d))^
(1/2)*x^2*a^2*b^2*e^4+1449*A*(b*(a*e-b*d))^(1/2)*x^2*b^4*d^2*e^2-280*B*(b*(a*e-b
*d))^(1/2)*x^2*a^3*b*e^4-644*B*(b*(a*e-b*d))^(1/2)*x^2*b^4*d^3*e+16*B*(b*(a*e-b*
d))^(1/2)*a^4*d*e^3-30*B*(b*(a*e-b*d))^(1/2)*a*b^3*d^4-168*A*(b*(a*e-b*d))^(1/2)
*a^3*b*d*e^3+864*A*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2+255*A*(b*(a*e-b*d))^(1/2)
*a*b^3*d^3*e-272*B*(b*(a*e-b*d))^(1/2)*a^3*b*d^2*e^2-659*B*(b*(a*e-b*d))^(1/2)*a
^2*b^2*d^3*e)*(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)

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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)/((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 = 0.314594, 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)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[1/120*(48*A*a^4*e^4 - 60*(B*a*b^3 + A*b^4)*d^4 - 2*(659*B*a^2*b^2 - 255*A*a*b^3
)*d^3*e - 32*(17*B*a^3*b - 54*A*a^2*b^2)*d^2*e^2 + 16*(2*B*a^4 - 21*A*a^3*b)*d*e
^3 - 210*(4*B*b^4*d*e^3 + (5*B*a*b^3 - 9*A*b^4)*e^4)*x^4 - 70*(28*B*b^4*d^2*e^2
+ (55*B*a*b^3 - 63*A*b^4)*d*e^3 + 5*(5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 - 14*(92*
B*b^4*d^3*e + 9*(39*B*a*b^3 - 23*A*b^4)*d^2*e^2 + 3*(109*B*a^2*b^2 - 177*A*a*b^3
)*d*e^3 + 8*(5*B*a^3*b - 9*A*a^2*b^2)*e^4)*x^2 + 105*(4*B*a^2*b^2*d^3*e + (5*B*a
^3*b - 9*A*a^2*b^2)*d^2*e^2 + (4*B*b^4*d*e^3 + (5*B*a*b^3 - 9*A*b^4)*e^4)*x^4 +
2*(4*B*b^4*d^2*e^2 + 9*(B*a*b^3 - A*b^4)*d*e^3 + (5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*
x^3 + (4*B*b^4*d^3*e + 3*(7*B*a*b^3 - 3*A*b^4)*d^2*e^2 + 12*(2*B*a^2*b^2 - 3*A*a
*b^3)*d*e^3 + (5*B*a^3*b - 9*A*a^2*b^2)*e^4)*x^2 + 2*(4*B*a*b^3*d^3*e + 9*(B*a^2
*b^2 - A*a*b^3)*d^2*e^2 + (5*B*a^3*b - 9*A*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)) - 2*(60*B*b^4*d^4 + (1183*B*a*b^3 - 135*A*b^4)*d^3*e + 3*
(643*B*a^2*b^2 - 831*A*a*b^3)*d^2*e^2 + 72*(9*B*a^3*b - 17*A*a^2*b^2)*d*e^3 - 8*
(5*B*a^4 - 9*A*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)), 1/60*(24*A*a^4*e^4 - 30*(
B*a*b^3 + A*b^4)*d^4 - (659*B*a^2*b^2 - 255*A*a*b^3)*d^3*e - 16*(17*B*a^3*b - 54
*A*a^2*b^2)*d^2*e^2 + 8*(2*B*a^4 - 21*A*a^3*b)*d*e^3 - 105*(4*B*b^4*d*e^3 + (5*B
*a*b^3 - 9*A*b^4)*e^4)*x^4 - 35*(28*B*b^4*d^2*e^2 + (55*B*a*b^3 - 63*A*b^4)*d*e^
3 + 5*(5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 - 7*(92*B*b^4*d^3*e + 9*(39*B*a*b^3 - 2
3*A*b^4)*d^2*e^2 + 3*(109*B*a^2*b^2 - 177*A*a*b^3)*d*e^3 + 8*(5*B*a^3*b - 9*A*a^
2*b^2)*e^4)*x^2 + 105*(4*B*a^2*b^2*d^3*e + (5*B*a^3*b - 9*A*a^2*b^2)*d^2*e^2 + (
4*B*b^4*d*e^3 + (5*B*a*b^3 - 9*A*b^4)*e^4)*x^4 + 2*(4*B*b^4*d^2*e^2 + 9*(B*a*b^3
- A*b^4)*d*e^3 + (5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 + (4*B*b^4*d^3*e + 3*(7*B*a
*b^3 - 3*A*b^4)*d^2*e^2 + 12*(2*B*a^2*b^2 - 3*A*a*b^3)*d*e^3 + (5*B*a^3*b - 9*A*
a^2*b^2)*e^4)*x^2 + 2*(4*B*a*b^3*d^3*e + 9*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (5*B*
a^3*b - 9*A*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)) - (60*B*b^4*d^4 + (1183*B*a*b^3 -
135*A*b^4)*d^3*e + 3*(643*B*a^2*b^2 - 831*A*a*b^3)*d^2*e^2 + 72*(9*B*a^3*b - 17*
A*a^2*b^2)*d*e^3 - 8*(5*B*a^4 - 9*A*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((B*x+A)/(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 = 0.356333, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done