3.2237 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=257 \[ \frac{5 \sqrt{e} (b d-a e) (-7 a B e+4 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{9/2}}+\frac{5 e \sqrt{a+b x} \sqrt{d+e x} (-7 a B e+4 A b e+3 b B d)}{4 b^4}+\frac{5 e \sqrt{a+b x} (d+e x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{6 b^3 (b d-a e)}-\frac{2 (d+e x)^{5/2} (-7 a B e+4 A b e+3 b B d)}{3 b^2 \sqrt{a+b x} (b d-a e)}-\frac{2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

[Out]

(5*e*(3*b*B*d + 4*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*b^4) + (5*e*(
3*b*B*d + 4*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(6*b^3*(b*d - a*e))
- (2*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(3*b^2*(b*d - a*e)*Sqrt[a +
b*x]) - (2*(A*b - a*B)*(d + e*x)^(7/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + (5*S
qrt[e]*(b*d - a*e)*(3*b*B*d + 4*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])
/(Sqrt[b]*Sqrt[d + e*x])])/(4*b^(9/2))

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Rubi [A]  time = 0.515561, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{5 \sqrt{e} (b d-a e) (-7 a B e+4 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{9/2}}+\frac{5 e \sqrt{a+b x} \sqrt{d+e x} (-7 a B e+4 A b e+3 b B d)}{4 b^4}+\frac{5 e \sqrt{a+b x} (d+e x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{6 b^3 (b d-a e)}-\frac{2 (d+e x)^{5/2} (-7 a B e+4 A b e+3 b B d)}{3 b^2 \sqrt{a+b x} (b d-a e)}-\frac{2 (d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(5/2),x]

[Out]

(5*e*(3*b*B*d + 4*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*b^4) + (5*e*(
3*b*B*d + 4*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(6*b^3*(b*d - a*e))
- (2*(3*b*B*d + 4*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(3*b^2*(b*d - a*e)*Sqrt[a +
b*x]) - (2*(A*b - a*B)*(d + e*x)^(7/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + (5*S
qrt[e]*(b*d - a*e)*(3*b*B*d + 4*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])
/(Sqrt[b]*Sqrt[d + e*x])])/(4*b^(9/2))

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Rubi in Sympy [A]  time = 49.0083, size = 253, normalized size = 0.98 \[ \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (a e - b d\right )} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (4 A b e - 7 B a e + 3 B b d\right )}{3 b^{2} \sqrt{a + b x} \left (a e - b d\right )} - \frac{5 e \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (4 A b e - 7 B a e + 3 B b d\right )}{6 b^{3} \left (a e - b d\right )} + \frac{5 e \sqrt{a + b x} \sqrt{d + e x} \left (4 A b e - 7 B a e + 3 B b d\right )}{4 b^{4}} - \frac{5 \sqrt{e} \left (a e - b d\right ) \left (4 A b e - 7 B a e + 3 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{4 b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(5/2)/(b*x+a)**(5/2),x)

[Out]

2*(d + e*x)**(7/2)*(A*b - B*a)/(3*b*(a + b*x)**(3/2)*(a*e - b*d)) + 2*(d + e*x)*
*(5/2)*(4*A*b*e - 7*B*a*e + 3*B*b*d)/(3*b**2*sqrt(a + b*x)*(a*e - b*d)) - 5*e*sq
rt(a + b*x)*(d + e*x)**(3/2)*(4*A*b*e - 7*B*a*e + 3*B*b*d)/(6*b**3*(a*e - b*d))
+ 5*e*sqrt(a + b*x)*sqrt(d + e*x)*(4*A*b*e - 7*B*a*e + 3*B*b*d)/(4*b**4) - 5*sqr
t(e)*(a*e - b*d)*(4*A*b*e - 7*B*a*e + 3*B*b*d)*atanh(sqrt(e)*sqrt(a + b*x)/(sqrt
(b)*sqrt(d + e*x)))/(4*b**(9/2))

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Mathematica [A]  time = 0.571996, size = 230, normalized size = 0.89 \[ \frac{\sqrt{d+e x} \left (4 A b \left (15 a^2 e^2-10 a b e (d-2 e x)+b^2 \left (-2 d^2-14 d e x+3 e^2 x^2\right )\right )+B \left (-105 a^3 e^2+5 a^2 b e (23 d-28 e x)+a b^2 \left (-16 d^2+158 d e x-21 e^2 x^2\right )+3 b^3 x \left (-8 d^2+9 d e x+2 e^2 x^2\right )\right )\right )}{12 b^4 (a+b x)^{3/2}}+\frac{5 \sqrt{e} (b d-a e) (-7 a B e+4 A b e+3 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{8 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(5/2),x]

[Out]

(Sqrt[d + e*x]*(B*(-105*a^3*e^2 + 5*a^2*b*e*(23*d - 28*e*x) + a*b^2*(-16*d^2 + 1
58*d*e*x - 21*e^2*x^2) + 3*b^3*x*(-8*d^2 + 9*d*e*x + 2*e^2*x^2)) + 4*A*b*(15*a^2
*e^2 - 10*a*b*e*(d - 2*e*x) + b^2*(-2*d^2 - 14*d*e*x + 3*e^2*x^2))))/(12*b^4*(a
+ b*x)^(3/2)) + (5*Sqrt[e]*(b*d - a*e)*(3*b*B*d + 4*A*b*e - 7*a*B*e)*Log[b*d + a
*e + 2*b*e*x + 2*Sqrt[b]*Sqrt[e]*Sqrt[a + b*x]*Sqrt[d + e*x]])/(8*b^(9/2))

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Maple [B]  time = 0.04, size = 1250, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^(5/2),x)

[Out]

-1/24*(e*x+d)^(1/2)*(-105*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2
)+a*e+b*d)/(b*e)^(1/2))*a^4*e^3-316*B*x*a*b^2*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^
(1/2)-24*A*x^2*b^3*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+48*B*x*b^3*d^2*((b*x+
a)*(e*x+d))^(1/2)*(b*e)^(1/2)-120*A*a^2*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2
)+32*B*a*b^2*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+120*A*ln(1/2*(2*b*x*e+2*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a^2*b^2*e^3-210*B*ln(1/
2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a^3*b*e
^3-60*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/
2))*a^2*b^2*d*e^2+150*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*
e+b*d)/(b*e)^(1/2))*a^3*b*d*e^2-45*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^2*d^2*e-12*B*x^3*b^3*e^2*((b*x+a)*(e*x+d)
)^(1/2)*(b*e)^(1/2)+16*A*b^3*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+210*B*a^3*e
^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+60*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b*e^3+60*A*ln(1/2*(2*b*x*e+2*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*a*b^3*e^3-60*A*ln(1/2*(2*
b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*b^4*d*e^2-
105*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2)
)*x^2*a^2*b^2*e^3-45*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e
+b*d)/(b*e)^(1/2))*x^2*b^4*d^2*e+150*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)
*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x^2*a*b^3*d*e^2+42*B*x^2*a*b^2*e^2*((b*x+a)*(
e*x+d))^(1/2)*(b*e)^(1/2)-54*B*x^2*b^3*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-1
60*A*x*a*b^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+112*A*x*b^3*d*e*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)+280*B*x*a^2*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+8
0*A*a*b^2*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-230*B*a^2*b*d*e*((b*x+a)*(e*x+
d))^(1/2)*(b*e)^(1/2)-120*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2
)+a*e+b*d)/(b*e)^(1/2))*x*a*b^3*d*e^2+300*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a^2*b^2*d*e^2-90*B*ln(1/2*(2*b*x*e+2*(
(b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a*b^3*d^2*e)/((b*x+a)
*(e*x+d))^(1/2)/(b*e)^(1/2)/(b*x+a)^(3/2)/b^4

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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)^(5/2)/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.14401, 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)^(5/2)/(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(15*(3*B*a^2*b^2*d^2 - 2*(5*B*a^3*b - 2*A*a^2*b^2)*d*e + (7*B*a^4 - 4*A*a^
3*b)*e^2 + (3*B*b^4*d^2 - 2*(5*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^2 - 4*A*a*b^3
)*e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d*e + (7*B*a^3*b - 4
*A*a^2*b^2)*e^2)*x)*sqrt(e/b)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2
+ 4*(2*b^2*e*x + b^2*d + a*b*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(e/b) + 8*(b^2*d
*e + a*b*e^2)*x) + 4*(6*B*b^3*e^2*x^3 - 8*(2*B*a*b^2 + A*b^3)*d^2 + 5*(23*B*a^2*
b - 8*A*a*b^2)*d*e - 15*(7*B*a^3 - 4*A*a^2*b)*e^2 + 3*(9*B*b^3*d*e - (7*B*a*b^2
- 4*A*b^3)*e^2)*x^2 - 2*(12*B*b^3*d^2 - (79*B*a*b^2 - 28*A*b^3)*d*e + 10*(7*B*a^
2*b - 4*A*a*b^2)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2
*b^4), 1/24*(15*(3*B*a^2*b^2*d^2 - 2*(5*B*a^3*b - 2*A*a^2*b^2)*d*e + (7*B*a^4 -
4*A*a^3*b)*e^2 + (3*B*b^4*d^2 - 2*(5*B*a*b^3 - 2*A*b^4)*d*e + (7*B*a^2*b^2 - 4*A
*a*b^3)*e^2)*x^2 + 2*(3*B*a*b^3*d^2 - 2*(5*B*a^2*b^2 - 2*A*a*b^3)*d*e + (7*B*a^3
*b - 4*A*a^2*b^2)*e^2)*x)*sqrt(-e/b)*arctan(1/2*(2*b*e*x + b*d + a*e)/(sqrt(b*x
+ a)*sqrt(e*x + d)*b*sqrt(-e/b))) + 2*(6*B*b^3*e^2*x^3 - 8*(2*B*a*b^2 + A*b^3)*d
^2 + 5*(23*B*a^2*b - 8*A*a*b^2)*d*e - 15*(7*B*a^3 - 4*A*a^2*b)*e^2 + 3*(9*B*b^3*
d*e - (7*B*a*b^2 - 4*A*b^3)*e^2)*x^2 - 2*(12*B*b^3*d^2 - (79*B*a*b^2 - 28*A*b^3)
*d*e + 10*(7*B*a^2*b - 4*A*a*b^2)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^6*x^2
+ 2*a*b^5*x + a^2*b^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)**(5/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.664108, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/(b*x + a)^(5/2),x, algorithm="giac")

[Out]

sage0*x