3.2228 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(a+b x)^{3/2}} \, dx\)
Optimal. Leaf size=249 \[ \frac{5 (b d-a e)^2 (-7 a B e+6 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{9/2} \sqrt{e}}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-7 a B e+6 A b e+b B d)}{8 b^4}+\frac{5 \sqrt{a+b x} (d+e x)^{3/2} (-7 a B e+6 A b e+b B d)}{12 b^3}+\frac{\sqrt{a+b x} (d+e x)^{5/2} (-7 a B e+6 A b e+b B d)}{3 b^2 (b d-a e)}-\frac{2 (d+e x)^{7/2} (A b-a B)}{b \sqrt{a+b x} (b d-a e)} \]
[Out]
(5*(b*d - a*e)*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^4)
+ (5*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^3) + ((b*B
*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b^2*(b*d - a*e)) - (2*
(A*b - a*B)*(d + e*x)^(7/2))/(b*(b*d - a*e)*Sqrt[a + b*x]) + (5*(b*d - a*e)^2*(b
*B*d + 6*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x]
)])/(8*b^(9/2)*Sqrt[e])
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Rubi [A] time = 0.518447, antiderivative size = 249, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167
\[ \frac{5 (b d-a e)^2 (-7 a B e+6 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{9/2} \sqrt{e}}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-7 a B e+6 A b e+b B d)}{8 b^4}+\frac{5 \sqrt{a+b x} (d+e x)^{3/2} (-7 a B e+6 A b e+b B d)}{12 b^3}+\frac{\sqrt{a+b x} (d+e x)^{5/2} (-7 a B e+6 A b e+b B d)}{3 b^2 (b d-a e)}-\frac{2 (d+e x)^{7/2} (A b-a B)}{b \sqrt{a+b x} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(3/2),x]
[Out]
(5*(b*d - a*e)*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^4)
+ (5*(b*B*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^3) + ((b*B
*d + 6*A*b*e - 7*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b^2*(b*d - a*e)) - (2*
(A*b - a*B)*(d + e*x)^(7/2))/(b*(b*d - a*e)*Sqrt[a + b*x]) + (5*(b*d - a*e)^2*(b
*B*d + 6*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x]
)])/(8*b^(9/2)*Sqrt[e])
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Rubi in Sympy [A] time = 50.0028, size = 241, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (A b - B a\right )}{b \sqrt{a + b x} \left (a e - b d\right )} - \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (6 A b e - 7 B a e + B b d\right )}{3 b^{2} \left (a e - b d\right )} + \frac{5 \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (6 A b e - 7 B a e + B b d\right )}{12 b^{3}} - \frac{5 \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right ) \left (6 A b e - 7 B a e + B b d\right )}{8 b^{4}} + \frac{5 \left (a e - b d\right )^{2} \left (6 A b e - 7 B a e + B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{8 b^{\frac{9}{2}} \sqrt{e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(5/2)/(b*x+a)**(3/2),x)
[Out]
2*(d + e*x)**(7/2)*(A*b - B*a)/(b*sqrt(a + b*x)*(a*e - b*d)) - sqrt(a + b*x)*(d
+ e*x)**(5/2)*(6*A*b*e - 7*B*a*e + B*b*d)/(3*b**2*(a*e - b*d)) + 5*sqrt(a + b*x)
*(d + e*x)**(3/2)*(6*A*b*e - 7*B*a*e + B*b*d)/(12*b**3) - 5*sqrt(a + b*x)*sqrt(d
+ e*x)*(a*e - b*d)*(6*A*b*e - 7*B*a*e + B*b*d)/(8*b**4) + 5*(a*e - b*d)**2*(6*A
*b*e - 7*B*a*e + B*b*d)*atanh(sqrt(e)*sqrt(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(8*
b**(9/2)*sqrt(e))
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Mathematica [A] time = 0.473828, size = 231, normalized size = 0.93 \[ \frac{\sqrt{d+e x} \left (B \left (105 a^3 e^2+5 a^2 b e (7 e x-38 d)+a b^2 \left (81 d^2-68 d e x-14 e^2 x^2\right )+b^3 x \left (33 d^2+26 d e x+8 e^2 x^2\right )\right )-6 A b \left (15 a^2 e^2+5 a b e (e x-5 d)+b^2 \left (8 d^2-9 d e x-2 e^2 x^2\right )\right )\right )}{24 b^4 \sqrt{a+b x}}+\frac{5 (b d-a e)^2 (-7 a B e+6 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{16 b^{9/2} \sqrt{e}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(5/2))/(a + b*x)^(3/2),x]
[Out]
(Sqrt[d + e*x]*(-6*A*b*(15*a^2*e^2 + 5*a*b*e*(-5*d + e*x) + b^2*(8*d^2 - 9*d*e*x
- 2*e^2*x^2)) + B*(105*a^3*e^2 + 5*a^2*b*e*(-38*d + 7*e*x) + a*b^2*(81*d^2 - 68
*d*e*x - 14*e^2*x^2) + b^3*x*(33*d^2 + 26*d*e*x + 8*e^2*x^2))))/(24*b^4*Sqrt[a +
b*x]) + (5*(b*d - a*e)^2*(b*B*d + 6*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]])/(16*b^(9/2)*Sqrt[e])
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Maple [B] time = 0.049, size = 1184, normalized size = 4.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^(3/2),x)
[Out]
1/48*(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^3-136*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)+66*B*x*b^3*d^2*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)-180*A*a^2*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)
+162*B*a*b^2*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+90*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+90*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*b^4*d^2*e
-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*a^3*b*e^3-180*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+90*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*b^3*d^2*e+225*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^2-135*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+16*B*x
^3*b^3*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-96*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)+15*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*b^4*d^3+90*
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+15*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*b^3*d^3-28*B*x^2*a*b^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+52*B*x
^2*b^3*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-60*A*x*a*b^2*e^2*((b*x+a)*(e*x+d)
)^(1/2)*(b*e)^(1/2)+108*A*x*b^3*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+70*B*x*a
^2*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+300*A*a*b^2*d*e*((b*x+a)*(e*x+d))^(
1/2)*(b*e)^(1/2)-380*B*a^2*b*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-180*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+225*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^2-135*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)^(1/2)/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)^(3/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.01547, 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)^(3/2),x, algorithm="fricas")
[Out]
[1/96*(4*(8*B*b^3*e^2*x^3 + 3*(27*B*a*b^2 - 16*A*b^3)*d^2 - 10*(19*B*a^2*b - 15*
A*a*b^2)*d*e + 15*(7*B*a^3 - 6*A*a^2*b)*e^2 + 2*(13*B*b^3*d*e - (7*B*a*b^2 - 6*A
*b^3)*e^2)*x^2 + (33*B*b^3*d^2 - 2*(34*B*a*b^2 - 27*A*b^3)*d*e + 5*(7*B*a^2*b -
6*A*a*b^2)*e^2)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) - 15*(B*a*b^3*d^3 - 3*(
3*B*a^2*b^2 - 2*A*a*b^3)*d^2*e + 3*(5*B*a^3*b - 4*A*a^2*b^2)*d*e^2 - (7*B*a^4 -
6*A*a^3*b)*e^3 + (B*b^4*d^3 - 3*(3*B*a*b^3 - 2*A*b^4)*d^2*e + 3*(5*B*a^2*b^2 - 4
*A*a*b^3)*d*e^2 - (7*B*a^3*b - 6*A*a^2*b^2)*e^3)*x)*log(-4*(2*b^2*e^2*x + b^2*d*
e + a*b*e^2)*sqrt(b*x + a)*sqrt(e*x + d) + (8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e
+ a^2*e^2 + 8*(b^2*d*e + a*b*e^2)*x)*sqrt(b*e)))/((b^5*x + a*b^4)*sqrt(b*e)), 1/
48*(2*(8*B*b^3*e^2*x^3 + 3*(27*B*a*b^2 - 16*A*b^3)*d^2 - 10*(19*B*a^2*b - 15*A*a
*b^2)*d*e + 15*(7*B*a^3 - 6*A*a^2*b)*e^2 + 2*(13*B*b^3*d*e - (7*B*a*b^2 - 6*A*b^
3)*e^2)*x^2 + (33*B*b^3*d^2 - 2*(34*B*a*b^2 - 27*A*b^3)*d*e + 5*(7*B*a^2*b - 6*A
*a*b^2)*e^2)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 15*(B*a*b^3*d^3 - 3*(3*
B*a^2*b^2 - 2*A*a*b^3)*d^2*e + 3*(5*B*a^3*b - 4*A*a^2*b^2)*d*e^2 - (7*B*a^4 - 6*
A*a^3*b)*e^3 + (B*b^4*d^3 - 3*(3*B*a*b^3 - 2*A*b^4)*d^2*e + 3*(5*B*a^2*b^2 - 4*A
*a*b^3)*d*e^2 - (7*B*a^3*b - 6*A*a^2*b^2)*e^3)*x)*arctan(1/2*(2*b*e*x + b*d + a*
e)*sqrt(-b*e)/(sqrt(b*x + a)*sqrt(e*x + d)*b*e)))/((b^5*x + a*b^4)*sqrt(-b*e))]
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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)**(3/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.658943, 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)^(3/2),x, algorithm="giac")
[Out]
sage0*x