3.2236 \(\int \frac{(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=302 \[ \frac{35 \sqrt{e} (b d-a e)^2 (-3 a B e+2 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{11/2}}+\frac{35 e \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^5}+\frac{35 e \sqrt{a+b x} (d+e x)^{3/2} (-3 a B e+2 A b e+b B d)}{12 b^4}+\frac{7 e \sqrt{a+b x} (d+e x)^{5/2} (-3 a B e+2 A b e+b B d)}{3 b^3 (b d-a e)}-\frac{2 (d+e x)^{7/2} (-3 a B e+2 A b e+b B d)}{b^2 \sqrt{a+b x} (b d-a e)}-\frac{2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

[Out]

(35*e*(b*d - a*e)*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^
5) + (35*e*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^4) +
(7*e*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b^3*(b*d - a
*e)) - (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)^(7/2))/(b^2*(b*d - a*e)*Sqrt[a +
b*x]) - (2*(A*b - a*B)*(d + e*x)^(9/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + (35
*Sqrt[e]*(b*d - a*e)^2*(b*B*d + 2*A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x
])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(11/2))

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Rubi [A]  time = 0.634484, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{35 \sqrt{e} (b d-a e)^2 (-3 a B e+2 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{11/2}}+\frac{35 e \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^5}+\frac{35 e \sqrt{a+b x} (d+e x)^{3/2} (-3 a B e+2 A b e+b B d)}{12 b^4}+\frac{7 e \sqrt{a+b x} (d+e x)^{5/2} (-3 a B e+2 A b e+b B d)}{3 b^3 (b d-a e)}-\frac{2 (d+e x)^{7/2} (-3 a B e+2 A b e+b B d)}{b^2 \sqrt{a+b x} (b d-a e)}-\frac{2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2),x]

[Out]

(35*e*(b*d - a*e)*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^
5) + (35*e*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^4) +
(7*e*(b*B*d + 2*A*b*e - 3*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b^3*(b*d - a
*e)) - (2*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)^(7/2))/(b^2*(b*d - a*e)*Sqrt[a +
b*x]) - (2*(A*b - a*B)*(d + e*x)^(9/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + (35
*Sqrt[e]*(b*d - a*e)^2*(b*B*d + 2*A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x
])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(11/2))

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Rubi in Sympy [A]  time = 63.837, size = 298, normalized size = 0.99 \[ \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (a e - b d\right )} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (2 A b e - 3 B a e + B b d\right )}{b^{2} \sqrt{a + b x} \left (a e - b d\right )} - \frac{7 e \sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (2 A b e - 3 B a e + B b d\right )}{3 b^{3} \left (a e - b d\right )} + \frac{35 e \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (2 A b e - 3 B a e + B b d\right )}{12 b^{4}} - \frac{35 e \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right ) \left (2 A b e - 3 B a e + B b d\right )}{8 b^{5}} + \frac{35 \sqrt{e} \left (a e - b d\right )^{2} \left (2 A b e - 3 B a e + B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{8 b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a)**(5/2),x)

[Out]

2*(d + e*x)**(9/2)*(A*b - B*a)/(3*b*(a + b*x)**(3/2)*(a*e - b*d)) + 2*(d + e*x)*
*(7/2)*(2*A*b*e - 3*B*a*e + B*b*d)/(b**2*sqrt(a + b*x)*(a*e - b*d)) - 7*e*sqrt(a
+ b*x)*(d + e*x)**(5/2)*(2*A*b*e - 3*B*a*e + B*b*d)/(3*b**3*(a*e - b*d)) + 35*e
*sqrt(a + b*x)*(d + e*x)**(3/2)*(2*A*b*e - 3*B*a*e + B*b*d)/(12*b**4) - 35*e*sqr
t(a + b*x)*sqrt(d + e*x)*(a*e - b*d)*(2*A*b*e - 3*B*a*e + B*b*d)/(8*b**5) + 35*s
qrt(e)*(a*e - b*d)**2*(2*A*b*e - 3*B*a*e + B*b*d)*atanh(sqrt(e)*sqrt(a + b*x)/(s
qrt(b)*sqrt(d + e*x)))/(8*b**(11/2))

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Mathematica [A]  time = 0.992664, size = 326, normalized size = 1.08 \[ \frac{\sqrt{d+e x} \left (B \left (315 a^4 e^3+210 a^3 b e^2 (2 e x-3 d)+7 a^2 b^2 e \left (49 d^2-122 d e x+9 e^2 x^2\right )-2 a b^3 \left (16 d^3-239 d^2 e x+69 d e^2 x^2+9 e^3 x^3\right )+b^4 x \left (-48 d^3+87 d^2 e x+38 d e^2 x^2+8 e^3 x^3\right )\right )-2 A b \left (105 a^3 e^3+35 a^2 b e^2 (4 e x-5 d)+7 a b^2 e \left (8 d^2-34 d e x+3 e^2 x^2\right )+b^3 \left (8 d^3+80 d^2 e x-39 d e^2 x^2-6 e^3 x^3\right )\right )\right )}{24 b^5 (a+b x)^{3/2}}+\frac{35 \sqrt{e} (b d-a e)^2 (-3 a B e+2 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{16 b^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^(5/2),x]

[Out]

(Sqrt[d + e*x]*(-2*A*b*(105*a^3*e^3 + 35*a^2*b*e^2*(-5*d + 4*e*x) + 7*a*b^2*e*(8
*d^2 - 34*d*e*x + 3*e^2*x^2) + b^3*(8*d^3 + 80*d^2*e*x - 39*d*e^2*x^2 - 6*e^3*x^
3)) + B*(315*a^4*e^3 + 210*a^3*b*e^2*(-3*d + 2*e*x) + 7*a^2*b^2*e*(49*d^2 - 122*
d*e*x + 9*e^2*x^2) + b^4*x*(-48*d^3 + 87*d^2*e*x + 38*d*e^2*x^2 + 8*e^3*x^3) - 2
*a*b^3*(16*d^3 - 239*d^2*e*x + 69*d*e^2*x^2 + 9*e^3*x^3))))/(24*b^5*(a + b*x)^(3
/2)) + (35*Sqrt[e]*(b*d - a*e)^2*(b*B*d + 2*A*b*e - 3*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]])/(16*b^(11/2))

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Maple [B]  time = 0.057, size = 1882, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^(5/2),x)

[Out]

1/48*(e*x+d)^(1/2)*(-276*B*x^2*a*b^3*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-3
2*A*b^4*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+952*A*x*a*b^3*d*e^2*((b*x+a)*(e*
x+d))^(1/2)*(b*e)^(1/2)-1708*B*x*a^2*b^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/
2)+956*B*x*a*b^3*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+210*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^3*d^2*e^2+735
*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*b*d*e^3-525*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^2*d^2*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))*a^2*b^3*d^3*e+16*B*x^4*b^4*e^3*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)+24*A*x^3*b^4*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-96*B*x*b^
4*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-420*A*a^3*b*e^3*((b*x+a)*(e*x+d))^(1/2
)*(b*e)^(1/2)-64*B*a*b^3*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+210*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^2*b^3*e
^4+210*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^5*d^2*e^2-315*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^3*b^2*e^4+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*b^5*d^3*e-315*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^5*e^4+630*B*a^4*e^3*(
(b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+210*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^4*b*e^4+420*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^3*b^2*e^4-630*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^4*b*e^4-420
*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^2*d*e^3-36*B*x^3*a*b^3*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+76*B*x^3*b^4
*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-84*A*x^2*a*b^3*e^3*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)+156*A*x^2*b^4*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+126*B*
x^2*a^2*b^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+174*B*x^2*b^4*d^2*e*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2)-560*A*x*a^2*b^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1
/2)-320*A*x*b^4*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+840*B*x*a^3*b*e^3*((b*
x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+700*A*a^2*b^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e
)^(1/2)-224*A*a*b^3*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-1260*B*a^3*b*d*e^2
*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+686*B*a^2*b^2*d^2*e*((b*x+a)*(e*x+d))^(1/2)
*(b*e)^(1/2)-420*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^4*d*e^3+735*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^3*d*e^3-525*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^4*d^2*e^2-840*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
^3*d*e^3+420*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^4*d^2*e^2+1470*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^2*d*e^3-1050*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^3*d^2*e^2+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*b^4*d^3
*e)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(b*x+a)^(3/2)/b^5

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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*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.79211, 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)^(5/2),x, algorithm="fricas")

[Out]

[-1/96*(105*(B*a^2*b^3*d^3 - (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e + (7*B*a^4*b - 4*
A*a^3*b^2)*d*e^2 - (3*B*a^5 - 2*A*a^4*b)*e^3 + (B*b^5*d^3 - (5*B*a*b^4 - 2*A*b^5
)*d^2*e + (7*B*a^2*b^3 - 4*A*a*b^4)*d*e^2 - (3*B*a^3*b^2 - 2*A*a^2*b^3)*e^3)*x^2
+ 2*(B*a*b^4*d^3 - (5*B*a^2*b^3 - 2*A*a*b^4)*d^2*e + (7*B*a^3*b^2 - 4*A*a^2*b^3
)*d*e^2 - (3*B*a^4*b - 2*A*a^3*b^2)*e^3)*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*(8*B*b^4*e^3*x^4 - 16*(2*B*a*b^3 +
A*b^4)*d^3 + 7*(49*B*a^2*b^2 - 16*A*a*b^3)*d^2*e - 70*(9*B*a^3*b - 5*A*a^2*b^2)*
d*e^2 + 105*(3*B*a^4 - 2*A*a^3*b)*e^3 + 2*(19*B*b^4*d*e^2 - 3*(3*B*a*b^3 - 2*A*b
^4)*e^3)*x^3 + 3*(29*B*b^4*d^2*e - 2*(23*B*a*b^3 - 13*A*b^4)*d*e^2 + 7*(3*B*a^2*
b^2 - 2*A*a*b^3)*e^3)*x^2 - 2*(24*B*b^4*d^3 - (239*B*a*b^3 - 80*A*b^4)*d^2*e + 7
*(61*B*a^2*b^2 - 34*A*a*b^3)*d*e^2 - 70*(3*B*a^3*b - 2*A*a^2*b^2)*e^3)*x)*sqrt(b
*x + a)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), 1/48*(105*(B*a^2*b^3*d^3
- (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e + (7*B*a^4*b - 4*A*a^3*b^2)*d*e^2 - (3*B*a^
5 - 2*A*a^4*b)*e^3 + (B*b^5*d^3 - (5*B*a*b^4 - 2*A*b^5)*d^2*e + (7*B*a^2*b^3 - 4
*A*a*b^4)*d*e^2 - (3*B*a^3*b^2 - 2*A*a^2*b^3)*e^3)*x^2 + 2*(B*a*b^4*d^3 - (5*B*a
^2*b^3 - 2*A*a*b^4)*d^2*e + (7*B*a^3*b^2 - 4*A*a^2*b^3)*d*e^2 - (3*B*a^4*b - 2*A
*a^3*b^2)*e^3)*x)*sqrt(-e/b)*arctan(1/2*(2*b*e*x + b*d + a*e)/(sqrt(b*x + a)*sqr
t(e*x + d)*b*sqrt(-e/b))) + 2*(8*B*b^4*e^3*x^4 - 16*(2*B*a*b^3 + A*b^4)*d^3 + 7*
(49*B*a^2*b^2 - 16*A*a*b^3)*d^2*e - 70*(9*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(3*
B*a^4 - 2*A*a^3*b)*e^3 + 2*(19*B*b^4*d*e^2 - 3*(3*B*a*b^3 - 2*A*b^4)*e^3)*x^3 +
3*(29*B*b^4*d^2*e - 2*(23*B*a*b^3 - 13*A*b^4)*d*e^2 + 7*(3*B*a^2*b^2 - 2*A*a*b^3
)*e^3)*x^2 - 2*(24*B*b^4*d^3 - (239*B*a*b^3 - 80*A*b^4)*d^2*e + 7*(61*B*a^2*b^2
- 34*A*a*b^3)*d*e^2 - 70*(3*B*a^3*b - 2*A*a^2*b^2)*e^3)*x)*sqrt(b*x + a)*sqrt(e*
x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*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*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.730737, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(b*x + a)^(5/2),x, algorithm="giac")

[Out]

sage0*x