3.2213 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}+\frac{2 b^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{9/2}}-\frac{2 b^2 B \sqrt{a+b x}}{e^4 \sqrt{d+e x}}-\frac{2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.266501, antiderivative size = 167, 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 \[ -\frac{2 (a+b x)^{7/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)}+\frac{2 b^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{9/2}}-\frac{2 b^2 B \sqrt{a+b x}}{e^4 \sqrt{d+e x}}-\frac{2 b B (a+b x)^{3/2}}{3 e^3 (d+e x)^{3/2}}-\frac{2 B (a+b x)^{5/2}}{5 e^2 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) - (2*B*(a + b
*x)^(5/2))/(5*e^2*(d + e*x)^(5/2)) - (2*b*B*(a + b*x)^(3/2))/(3*e^3*(d + e*x)^(3
/2)) - (2*b^2*B*Sqrt[a + b*x])/(e^4*Sqrt[d + e*x]) + (2*b^(5/2)*B*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 = 27.841, size = 156, normalized size = 0.93 \[ \frac{2 B b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{e^{\frac{9}{2}}} - \frac{2 B b^{2} \sqrt{a + b x}}{e^{4} \sqrt{d + e x}} - \frac{2 B b \left (a + b x\right )^{\frac{3}{2}}}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{7 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(9/2),x)

[Out]

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

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Mathematica [A]  time = 0.643495, size = 196, normalized size = 1.17 \[ \frac{2 \sqrt{a+b x} \left (-\frac{b^2 (d+e x)^3 (161 a B e+15 A b e-176 b B d)}{a e-b d}+b (d+e x)^2 (-77 a B e-45 A b e+122 b B d)-3 (d+e x) (a e-b d) (7 a B e+15 A b e-22 b B d)+15 (b d-a e)^2 (B d-A e)\right )}{105 e^4 (d+e x)^{7/2}}+\frac{b^{5/2} B \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{e^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(2*Sqrt[a + b*x]*(15*(b*d - a*e)^2*(B*d - A*e) - 3*(-(b*d) + a*e)*(-22*b*B*d + 1
5*A*b*e + 7*a*B*e)*(d + e*x) + b*(122*b*B*d - 45*A*b*e - 77*a*B*e)*(d + e*x)^2 -
(b^2*(-176*b*B*d + 15*A*b*e + 161*a*B*e)*(d + e*x)^3)/(-(b*d) + a*e)))/(105*e^4
*(d + e*x)^(7/2)) + (b^(5/2)*B*Log[b*d + a*e + 2*b*e*x + 2*Sqrt[b]*Sqrt[e]*Sqrt[
a + b*x]*Sqrt[d + e*x]])/e^(9/2)

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Maple [B]  time = 0.04, size = 1089, normalized size = 6.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(9/2),x)

[Out]

-1/105*(b*x+a)^(1/2)*(30*A*x^3*b^3*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+568*B
*x^2*a*b^2*d*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+322*B*x^3*a*b^2*e^4*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)-352*B*x^3*b^3*d*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2)+90*A*x^2*a*b^2*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+154*B*x^2*a^2*b*e^4*
((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-812*B*x^2*b^3*d^2*e^2*((b*x+a)*(e*x+d))^(1/2
)*(b*e)^(1/2)+90*A*x*a^2*b*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-700*B*x*b^3*d
^3*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+28*B*a^2*b*d^2*e^2*((b*x+a)*(e*x+d))^(1
/2)*(b*e)^(1/2)+140*B*a*b^2*d^3*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-420*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^
3*d*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^2*a*b^3*d^2*e^3-420*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^3*e^2+30*A*a^3*e^4*((b*x+a)*(e*x+d))^(1/
2)*(b*e)^(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^4*d^5+42*B*x*a^3*e^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+12*B
*a^3*d*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(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))*x^4*a*b^3*e^5+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^4*b^4*d*e^4+4
20*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^4*d^2*e^3+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^2*b^4*d^3*e^2+420*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^4*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))*a*b^3*d^4*e-210*B*b^3*d^4*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+92*B*x*a^2*b*d*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e
)^(1/2)+476*B*x*a*b^2*d^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/((b*x+a)*(e*x
+d))^(1/2)/(a*e-b*d)/(b*e)^(1/2)/(e*x+d)^(7/2)/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)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74188, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

[1/210*(105*(B*b^3*d^5 - B*a*b^2*d^4*e + (B*b^3*d*e^4 - B*a*b^2*e^5)*x^4 + 4*(B*
b^3*d^2*e^3 - B*a*b^2*d*e^4)*x^3 + 6*(B*b^3*d^3*e^2 - B*a*b^2*d^2*e^3)*x^2 + 4*(
B*b^3*d^4*e - B*a*b^2*d^3*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*(105*B*b^3*d^4 - 70*B*a*b^2*d^3*e - 14*B*a^2*
b*d^2*e^2 - 6*B*a^3*d*e^3 - 15*A*a^3*e^4 + (176*B*b^3*d*e^3 - (161*B*a*b^2 + 15*
A*b^3)*e^4)*x^3 + (406*B*b^3*d^2*e^2 - 284*B*a*b^2*d*e^3 - (77*B*a^2*b + 45*A*a*
b^2)*e^4)*x^2 + (350*B*b^3*d^3*e - 238*B*a*b^2*d^2*e^2 - 46*B*a^2*b*d*e^3 - 3*(7
*B*a^3 + 15*A*a^2*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*d^5*e^4 - a*d^4*e^5
+ (b*d*e^8 - a*e^9)*x^4 + 4*(b*d^2*e^7 - a*d*e^8)*x^3 + 6*(b*d^3*e^6 - a*d^2*e^
7)*x^2 + 4*(b*d^4*e^5 - a*d^3*e^6)*x), 1/105*(105*(B*b^3*d^5 - B*a*b^2*d^4*e + (
B*b^3*d*e^4 - B*a*b^2*e^5)*x^4 + 4*(B*b^3*d^2*e^3 - B*a*b^2*d*e^4)*x^3 + 6*(B*b^
3*d^3*e^2 - B*a*b^2*d^2*e^3)*x^2 + 4*(B*b^3*d^4*e - B*a*b^2*d^3*e^2)*x)*sqrt(-b/
e)*arctan(1/2*(2*b*e*x + b*d + a*e)/(sqrt(b*x + a)*sqrt(e*x + d)*e*sqrt(-b/e)))
- 2*(105*B*b^3*d^4 - 70*B*a*b^2*d^3*e - 14*B*a^2*b*d^2*e^2 - 6*B*a^3*d*e^3 - 15*
A*a^3*e^4 + (176*B*b^3*d*e^3 - (161*B*a*b^2 + 15*A*b^3)*e^4)*x^3 + (406*B*b^3*d^
2*e^2 - 284*B*a*b^2*d*e^3 - (77*B*a^2*b + 45*A*a*b^2)*e^4)*x^2 + (350*B*b^3*d^3*
e - 238*B*a*b^2*d^2*e^2 - 46*B*a^2*b*d*e^3 - 3*(7*B*a^3 + 15*A*a^2*b)*e^4)*x)*sq
rt(b*x + a)*sqrt(e*x + d))/(b*d^5*e^4 - a*d^4*e^5 + (b*d*e^8 - a*e^9)*x^4 + 4*(b
*d^2*e^7 - a*d*e^8)*x^3 + 6*(b*d^3*e^6 - a*d^2*e^7)*x^2 + 4*(b*d^4*e^5 - a*d^3*e
^6)*x)]

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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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.3382, size = 946, normalized size = 5.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

1/768*B*sqrt(b)*abs(b)*e^(1/2)*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*
d + (b*x + a)*b*e - a*b*e)))/(b^10*d*e^6 - a*b^9*e^7) + 1/80640*(((b*x + a)*((17
6*B*b^10*d^3*abs(b)*e^6 - 513*B*a*b^9*d^2*abs(b)*e^7 - 15*A*b^10*d^2*abs(b)*e^7
+ 498*B*a^2*b^8*d*abs(b)*e^8 + 30*A*a*b^9*d*abs(b)*e^8 - 161*B*a^3*b^7*abs(b)*e^
9 - 15*A*a^2*b^8*abs(b)*e^9)*(b*x + a)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*
b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12) + 406*(B*b^11*d^4*abs(b)*e^5
- 4*B*a*b^10*d^3*abs(b)*e^6 + 6*B*a^2*b^9*d^2*abs(b)*e^7 - 4*B*a^3*b^8*d*abs(b)*
e^8 + B*a^4*b^7*abs(b)*e^9)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d^2*e^
10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12)) + 350*(B*b^12*d^5*abs(b)*e^4 - 5*B*a*b^
11*d^4*abs(b)*e^5 + 10*B*a^2*b^10*d^3*abs(b)*e^6 - 10*B*a^3*b^9*d^2*abs(b)*e^7 +
5*B*a^4*b^8*d*abs(b)*e^8 - B*a^5*b^7*abs(b)*e^9)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e
^9 + 6*a^2*b^14*d^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12))*(b*x + a) + 105*(
B*b^13*d^6*abs(b)*e^3 - 6*B*a*b^12*d^5*abs(b)*e^4 + 15*B*a^2*b^11*d^4*abs(b)*e^5
- 20*B*a^3*b^10*d^3*abs(b)*e^6 + 15*B*a^4*b^9*d^2*abs(b)*e^7 - 6*B*a^5*b^8*d*ab
s(b)*e^8 + B*a^6*b^7*abs(b)*e^9)/(b^16*d^4*e^8 - 4*a*b^15*d^3*e^9 + 6*a^2*b^14*d
^2*e^10 - 4*a^3*b^13*d*e^11 + a^4*b^12*e^12))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b
*e - a*b*e)^(7/2)