3.2212 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx\)
Optimal. Leaf size=256 \[ -\frac{b^{3/2} (-5 a B e-2 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{9/2}}+\frac{b^2 \sqrt{a+b x} \sqrt{d+e x} (-5 a B e-2 A b e+7 b B d)}{e^4 (b d-a e)}-\frac{2 b (a+b x)^{3/2} (-5 a B e-2 A b e+7 b B d)}{3 e^3 \sqrt{d+e x} (b d-a e)}-\frac{2 (a+b x)^{5/2} (-5 a B e-2 A b e+7 b B d)}{15 e^2 (d+e x)^{3/2} (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) - (2*(7*b*B*d
- 2*A*b*e - 5*a*B*e)*(a + b*x)^(5/2))/(15*e^2*(b*d - a*e)*(d + e*x)^(3/2)) - (2
*b*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(a + b*x)^(3/2))/(3*e^3*(b*d - a*e)*Sqrt[d + e*
x]) + (b^2*(7*b*B*d - 2*A*b*e - 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(e^4*(b*d
- a*e)) - (b^(3/2)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])
/(Sqrt[b]*Sqrt[d + e*x])])/e^(9/2)
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Rubi [A] time = 0.490669, antiderivative size = 256, 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
\[ -\frac{b^{3/2} (-5 a B e-2 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{9/2}}+\frac{b^2 \sqrt{a+b x} \sqrt{d+e x} (-5 a B e-2 A b e+7 b B d)}{e^4 (b d-a e)}-\frac{2 b (a+b x)^{3/2} (-5 a B e-2 A b e+7 b B d)}{3 e^3 \sqrt{d+e x} (b d-a e)}-\frac{2 (a+b x)^{5/2} (-5 a B e-2 A b e+7 b B d)}{15 e^2 (d+e x)^{3/2} (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(7/2),x]
[Out]
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) - (2*(7*b*B*d
- 2*A*b*e - 5*a*B*e)*(a + b*x)^(5/2))/(15*e^2*(b*d - a*e)*(d + e*x)^(3/2)) - (2
*b*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(a + b*x)^(3/2))/(3*e^3*(b*d - a*e)*Sqrt[d + e*
x]) + (b^2*(7*b*B*d - 2*A*b*e - 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(e^4*(b*d
- a*e)) - (b^(3/2)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])
/(Sqrt[b]*Sqrt[d + e*x])])/e^(9/2)
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Rubi in Sympy [A] time = 46.2664, size = 246, normalized size = 0.96 \[ \frac{2 b^{\frac{3}{2}} \left (A b e + \frac{B \left (5 a e - 7 b d\right )}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{e^{\frac{9}{2}}} + \frac{2 b^{2} \sqrt{a + b x} \sqrt{d + e x} \left (A b e + \frac{B \left (5 a e - 7 b d\right )}{2}\right )}{e^{4} \left (a e - b d\right )} - \frac{2 b \left (a + b x\right )^{\frac{3}{2}} \left (2 A b e + 5 B a e - 7 B b d\right )}{3 e^{3} \sqrt{d + e x} \left (a e - b d\right )} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} - \frac{4 \left (a + b x\right )^{\frac{5}{2}} \left (A b e + \frac{B \left (5 a e - 7 b d\right )}{2}\right )}{15 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(7/2),x)
[Out]
2*b**(3/2)*(A*b*e + B*(5*a*e - 7*b*d)/2)*atanh(sqrt(b)*sqrt(d + e*x)/(sqrt(e)*sq
rt(a + b*x)))/e**(9/2) + 2*b**2*sqrt(a + b*x)*sqrt(d + e*x)*(A*b*e + B*(5*a*e -
7*b*d)/2)/(e**4*(a*e - b*d)) - 2*b*(a + b*x)**(3/2)*(2*A*b*e + 5*B*a*e - 7*B*b*d
)/(3*e**3*sqrt(d + e*x)*(a*e - b*d)) - 2*(a + b*x)**(7/2)*(A*e - B*d)/(5*e*(d +
e*x)**(5/2)*(a*e - b*d)) - 4*(a + b*x)**(5/2)*(A*b*e + B*(5*a*e - 7*b*d)/2)/(15*
e**2*(d + e*x)**(3/2)*(a*e - b*d))
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Mathematica [A] time = 0.510478, size = 190, normalized size = 0.74 \[ \frac{b^{3/2} (5 a B e+2 A b e-7 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{2 e^{9/2}}+\frac{\sqrt{a+b x} \left (2 b (d+e x)^2 (-35 a B e-23 A b e+58 b B d)-2 (d+e x) (b d-a e) (-5 a B e-11 A b e+16 b B d)+6 (b d-a e)^2 (B d-A e)+15 b^2 B (d+e x)^3\right )}{15 e^4 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(7/2),x]
[Out]
(Sqrt[a + b*x]*(6*(b*d - a*e)^2*(B*d - A*e) - 2*(b*d - a*e)*(16*b*B*d - 11*A*b*e
- 5*a*B*e)*(d + e*x) + 2*b*(58*b*B*d - 23*A*b*e - 35*a*B*e)*(d + e*x)^2 + 15*b^
2*B*(d + e*x)^3))/(15*e^4*(d + e*x)^(5/2)) + (b^(3/2)*(-7*b*B*d + 2*A*b*e + 5*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]])/(
2*e^(9/2))
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Maple [B] time = 0.037, size = 1092, normalized size = 4.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(7/2),x)
[Out]
1/30*(-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^3*b^3*d*e^3-315*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*b^3*d^2*e^2-92*A*x^2*b^2*e^3*(b*e)^(1/2)*((b*x+a)*(
e*x+d))^(1/2)-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*b^3*d^3*e+75*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^2*d^3*e-20*B*x*a^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*
x+d))^(1/2)-8*B*a^2*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+210*B*b^2*d^3*(b*e
)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+30*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^3*b^3*e^4+75*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^3*a*b^2*e^4+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^2*a*b^2*d*e^3+22
5*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^2*d^2*e^2-140*B*x^2*a*b*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+322*B*x^2*
b^2*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-44*A*x*a*b*e^3*(b*e)^(1/2)*((b*x+a
)*(e*x+d))^(1/2)+490*B*x*b^2*d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-80*B*a*b*
d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-196*B*x*a*b*d*e^2*(b*e)^(1/2)*((b*x+a)
*(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))*b^3*d^4-12*A*a^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-60*A*
b^2*d^2*e*((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*b^3*d^2*e^2+30*B*x^3*b^2*e^3*((
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^2*b^3*d*e^3-140*A*x*b^2*d*e^2*((b*x+a)*(e*x
+d))^(1/2)*(b*e)^(1/2)-20*A*a*b*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+30*A*l
n(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))*b^3*d
^3*e)*(b*x+a)^(1/2)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(e*x+d)^(5/2)/e^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)*(b*x + a)^(5/2)/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.31493, 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*x + a)^(5/2)/(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
[-1/60*(15*(7*B*b^2*d^4 - (5*B*a*b + 2*A*b^2)*d^3*e + (7*B*b^2*d*e^3 - (5*B*a*b
+ 2*A*b^2)*e^4)*x^3 + 3*(7*B*b^2*d^2*e^2 - (5*B*a*b + 2*A*b^2)*d*e^3)*x^2 + 3*(7
*B*b^2*d^3*e - (5*B*a*b + 2*A*b^2)*d^2*e^2)*x)*sqrt(b/e)*log(8*b^2*e^2*x^2 + b^2
*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e^2*x + b*d*e + a*e^2)*sqrt(b*x + a)*sqrt(e*
x + d)*sqrt(b/e) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(15*B*b^2*e^3*x^3 + 105*B*b^2*d^
3 - 6*A*a^2*e^3 - 10*(4*B*a*b + 3*A*b^2)*d^2*e - 2*(2*B*a^2 + 5*A*a*b)*d*e^2 + (
161*B*b^2*d*e^2 - 2*(35*B*a*b + 23*A*b^2)*e^3)*x^2 + (245*B*b^2*d^2*e - 14*(7*B*
a*b + 5*A*b^2)*d*e^2 - 2*(5*B*a^2 + 11*A*a*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d
))/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4), -1/30*(15*(7*B*b^2*d^4 - (5*
B*a*b + 2*A*b^2)*d^3*e + (7*B*b^2*d*e^3 - (5*B*a*b + 2*A*b^2)*e^4)*x^3 + 3*(7*B*
b^2*d^2*e^2 - (5*B*a*b + 2*A*b^2)*d*e^3)*x^2 + 3*(7*B*b^2*d^3*e - (5*B*a*b + 2*A
*b^2)*d^2*e^2)*x)*sqrt(-b/e)*arctan(1/2*(2*b*e*x + b*d + a*e)/(sqrt(b*x + a)*sqr
t(e*x + d)*e*sqrt(-b/e))) - 2*(15*B*b^2*e^3*x^3 + 105*B*b^2*d^3 - 6*A*a^2*e^3 -
10*(4*B*a*b + 3*A*b^2)*d^2*e - 2*(2*B*a^2 + 5*A*a*b)*d*e^2 + (161*B*b^2*d*e^2 -
2*(35*B*a*b + 23*A*b^2)*e^3)*x^2 + (245*B*b^2*d^2*e - 14*(7*B*a*b + 5*A*b^2)*d*e
^2 - 2*(5*B*a^2 + 11*A*a*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^7*x^3 + 3*d*
e^6*x^2 + 3*d^2*e^5*x + d^3*e^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)**(5/2)*(B*x+A)/(e*x+d)**(7/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.307483, size = 902, normalized size = 3.52 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(7/2),x, algorithm="giac")
[Out]
(7*B*b^2*d*abs(b) - 5*B*a*b*abs(b)*e - 2*A*b^2*abs(b)*e)*e^(-9/2)*ln(abs(-sqrt(b
*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) + 1/15*(
((b*x + a)*(15*(B*b^9*d^2*abs(b)*e^6 - 2*B*a*b^8*d*abs(b)*e^7 + B*a^2*b^7*abs(b)
*e^8)*(b*x + a)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9) + 23*(7*B*b^10*d^3*a
bs(b)*e^5 - 19*B*a*b^9*d^2*abs(b)*e^6 - 2*A*b^10*d^2*abs(b)*e^6 + 17*B*a^2*b^8*d
*abs(b)*e^7 + 4*A*a*b^9*d*abs(b)*e^7 - 5*B*a^3*b^7*abs(b)*e^8 - 2*A*a^2*b^8*abs(
b)*e^8)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9)) + 35*(7*B*b^11*d^4*abs(b)*e
^4 - 26*B*a*b^10*d^3*abs(b)*e^5 - 2*A*b^11*d^3*abs(b)*e^5 + 36*B*a^2*b^9*d^2*abs
(b)*e^6 + 6*A*a*b^10*d^2*abs(b)*e^6 - 22*B*a^3*b^8*d*abs(b)*e^7 - 6*A*a^2*b^9*d*
abs(b)*e^7 + 5*B*a^4*b^7*abs(b)*e^8 + 2*A*a^3*b^8*abs(b)*e^8)/(b^6*d^2*e^7 - 2*a
*b^5*d*e^8 + a^2*b^4*e^9))*(b*x + a) + 15*(7*B*b^12*d^5*abs(b)*e^3 - 33*B*a*b^11
*d^4*abs(b)*e^4 - 2*A*b^12*d^4*abs(b)*e^4 + 62*B*a^2*b^10*d^3*abs(b)*e^5 + 8*A*a
*b^11*d^3*abs(b)*e^5 - 58*B*a^3*b^9*d^2*abs(b)*e^6 - 12*A*a^2*b^10*d^2*abs(b)*e^
6 + 27*B*a^4*b^8*d*abs(b)*e^7 + 8*A*a^3*b^9*d*abs(b)*e^7 - 5*B*a^5*b^7*abs(b)*e^
8 - 2*A*a^4*b^8*abs(b)*e^8)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9))*sqrt(b*
x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2)