3.1781 \(\int \frac{A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=302 \[ -\frac{e^3 (a+b x) \log (a+b x) (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e^3 (a+b x) (B d-A e) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{e^2 (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{e (B d-A e)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{B d-A e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.621492, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{e^3 (a+b x) \log (a+b x) (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e^3 (a+b x) (B d-A e) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{e^2 (B d-A e)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{e (B d-A e)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{B d-A e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 84.2187, size = 270, normalized size = 0.89 \[ - \frac{e^{3} \left (A e - B d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{e^{3} \left (A e - B d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{e^{2} \left (A e - B d\right )}{\left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{e \left (2 a + 2 b x\right ) \left (A e - B d\right )}{4 \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{A e - B d}{3 \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right )}{8 b \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**3*(A*e - B*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/((a + b*x)*(a*e
- b*d)**5) + e**3*(A*e - B*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/((a
+ b*x)*(a*e - b*d)**5) + e**2*(A*e - B*d)/((a*e - b*d)**4*sqrt(a**2 + 2*a*b*x +
b**2*x**2)) + e*(2*a + 2*b*x)*(A*e - B*d)/(4*(a*e - b*d)**3*(a**2 + 2*a*b*x + b*
*2*x**2)**(3/2)) + (A*e - B*d)/(3*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(
3/2)) + (2*a + 2*b*x)*(A*b - B*a)/(8*b*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)*
*(5/2))

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Mathematica [A]  time = 0.248686, size = 182, normalized size = 0.6 \[ \frac{12 e^3 (a+b x)^3 \log (a+b x) (A e-B d)+12 e^3 (a+b x)^3 (B d-A e) \log (d+e x)+12 e^2 (a+b x)^2 (b d-a e) (A e-B d)+\frac{3 (a B-A b) (b d-a e)^4}{b (a+b x)}-6 e (a+b x) (b d-a e)^2 (A e-B d)+4 (b d-a e)^3 (A e-B d)}{12 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \]

Antiderivative was successfully verified.

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

[Out]

(4*(b*d - a*e)^3*(-(B*d) + A*e) + (3*(-(A*b) + a*B)*(b*d - a*e)^4)/(b*(a + b*x))
- 6*e*(b*d - a*e)^2*(-(B*d) + A*e)*(a + b*x) + 12*e^2*(b*d - a*e)*(-(B*d) + A*e
)*(a + b*x)^2 + 12*e^3*(-(B*d) + A*e)*(a + b*x)^3*Log[a + b*x] + 12*e^3*(B*d - A
*e)*(a + b*x)^3*Log[d + e*x])/(12*(b*d - a*e)^5*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.033, size = 777, normalized size = 2.6 \[ -{\frac{ \left ( 12\,A\ln \left ( bx+a \right ){a}^{4}b{e}^{4}-52\,Ax{a}^{3}{b}^{2}{e}^{4}-12\,A{x}^{3}a{b}^{4}{e}^{4}-3\,A{b}^{5}{d}^{4}-72\,B\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{3}-48\,B\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}d{e}^{3}+6\,B{x}^{2}{b}^{5}{d}^{3}e+12\,A\ln \left ( bx+a \right ){x}^{4}{b}^{5}{e}^{4}-6\,A{x}^{2}{b}^{5}{d}^{2}{e}^{2}+72\,Ax{a}^{2}{b}^{3}d{e}^{3}-25\,A{a}^{4}b{e}^{4}+16\,Aa{b}^{4}{d}^{3}e-Ba{b}^{4}{d}^{4}+6\,B{a}^{2}{b}^{3}{d}^{3}e-4\,Bx{b}^{5}{d}^{4}+3\,B{a}^{5}{e}^{4}+12\,A{x}^{3}{b}^{5}d{e}^{3}-12\,B{x}^{3}{b}^{5}{d}^{2}{e}^{2}-42\,A{x}^{2}{a}^{2}{b}^{3}{e}^{4}+4\,Ax{b}^{5}{d}^{3}e-72\,A\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{3}{e}^{4}-48\,A\ln \left ( ex+d \right ) x{a}^{3}{b}^{2}{e}^{4}+12\,B\ln \left ( ex+d \right ){a}^{4}bd{e}^{3}+12\,B\ln \left ( ex+d \right ){x}^{4}{b}^{5}d{e}^{3}-48\,A\ln \left ( ex+d \right ){x}^{3}a{b}^{4}{e}^{4}-12\,B\ln \left ( bx+a \right ){x}^{4}{b}^{5}d{e}^{3}+48\,A\ln \left ( bx+a \right ){x}^{3}a{b}^{4}{e}^{4}+48\,B\ln \left ( ex+d \right ) x{a}^{3}{b}^{2}d{e}^{3}+48\,B\ln \left ( ex+d \right ){x}^{3}a{b}^{4}d{e}^{3}+72\,B\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{3}+12\,B{x}^{3}a{b}^{4}d{e}^{3}+48\,A{x}^{2}a{b}^{4}d{e}^{3}-24\,Axa{b}^{4}{d}^{2}{e}^{2}+24\,Bxa{b}^{4}{d}^{3}e+72\,A\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}{e}^{4}+48\,A\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}{e}^{4}-12\,B\ln \left ( bx+a \right ){a}^{4}bd{e}^{3}+52\,Bx{a}^{3}{b}^{2}d{e}^{3}-72\,Bx{a}^{2}{b}^{3}{d}^{2}{e}^{2}+42\,B{x}^{2}{a}^{2}{b}^{3}d{e}^{3}-48\,B{x}^{2}a{b}^{4}{d}^{2}{e}^{2}-12\,A\ln \left ( ex+d \right ){x}^{4}{b}^{5}{e}^{4}-12\,A\ln \left ( ex+d \right ){a}^{4}b{e}^{4}-18\,B{a}^{3}{b}^{2}{d}^{2}{e}^{2}+48\,A{a}^{3}{b}^{2}d{e}^{3}+10\,B{a}^{4}bd{e}^{3}-48\,B\ln \left ( bx+a \right ){x}^{3}a{b}^{4}d{e}^{3}-36\,A{a}^{2}{b}^{3}{d}^{2}{e}^{2} \right ) \left ( bx+a \right ) }{12\, \left ( ae-bd \right ) ^{5}b} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(12*A*ln(b*x+a)*a^4*b*e^4-52*A*x*a^3*b^2*e^4-12*A*x^3*a*b^4*e^4-3*A*b^5*d^
4-72*B*ln(b*x+a)*x^2*a^2*b^3*d*e^3-48*B*ln(b*x+a)*x*a^3*b^2*d*e^3+6*B*x^2*b^5*d^
3*e+12*A*ln(b*x+a)*x^4*b^5*e^4-6*A*x^2*b^5*d^2*e^2+72*A*x*a^2*b^3*d*e^3-25*A*a^4
*b*e^4+16*A*a*b^4*d^3*e-B*a*b^4*d^4+6*B*a^2*b^3*d^3*e-4*B*x*b^5*d^4+3*B*a^5*e^4+
12*A*x^3*b^5*d*e^3-12*B*x^3*b^5*d^2*e^2-42*A*x^2*a^2*b^3*e^4+4*A*x*b^5*d^3*e-72*
A*ln(e*x+d)*x^2*a^2*b^3*e^4-48*A*ln(e*x+d)*x*a^3*b^2*e^4+12*B*ln(e*x+d)*a^4*b*d*
e^3+12*B*ln(e*x+d)*x^4*b^5*d*e^3-48*A*ln(e*x+d)*x^3*a*b^4*e^4-12*B*ln(b*x+a)*x^4
*b^5*d*e^3+48*A*ln(b*x+a)*x^3*a*b^4*e^4+48*B*ln(e*x+d)*x*a^3*b^2*d*e^3+48*B*ln(e
*x+d)*x^3*a*b^4*d*e^3+72*B*ln(e*x+d)*x^2*a^2*b^3*d*e^3+12*B*x^3*a*b^4*d*e^3+48*A
*x^2*a*b^4*d*e^3-24*A*x*a*b^4*d^2*e^2+24*B*x*a*b^4*d^3*e+72*A*ln(b*x+a)*x^2*a^2*
b^3*e^4+48*A*ln(b*x+a)*x*a^3*b^2*e^4-12*B*ln(b*x+a)*a^4*b*d*e^3+52*B*x*a^3*b^2*d
*e^3-72*B*x*a^2*b^3*d^2*e^2+42*B*x^2*a^2*b^3*d*e^3-48*B*x^2*a*b^4*d^2*e^2-12*A*l
n(e*x+d)*x^4*b^5*e^4-12*A*ln(e*x+d)*a^4*b*e^4-18*B*a^3*b^2*d^2*e^2+48*A*a^3*b^2*
d*e^3+10*B*a^4*b*d*e^3-48*B*ln(b*x+a)*x^3*a*b^4*d*e^3-36*A*a^2*b^3*d^2*e^2)*(b*x
+a)/b/(a*e-b*d)^5/((b*x+a)^2)^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306054, size = 1308, normalized size = 4.33 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*((B*a*b^4 + 3*A*b^5)*d^4 - 2*(3*B*a^2*b^3 + 8*A*a*b^4)*d^3*e + 18*(B*a^3*b
^2 + 2*A*a^2*b^3)*d^2*e^2 - 2*(5*B*a^4*b + 24*A*a^3*b^2)*d*e^3 - (3*B*a^5 - 25*A
*a^4*b)*e^4 + 12*(B*b^5*d^2*e^2 + A*a*b^4*e^4 - (B*a*b^4 + A*b^5)*d*e^3)*x^3 - 6
*(B*b^5*d^3*e - 7*A*a^2*b^3*e^4 - (8*B*a*b^4 + A*b^5)*d^2*e^2 + (7*B*a^2*b^3 + 8
*A*a*b^4)*d*e^3)*x^2 + 4*(B*b^5*d^4 + 13*A*a^3*b^2*e^4 - (6*B*a*b^4 + A*b^5)*d^3
*e + 6*(3*B*a^2*b^3 + A*a*b^4)*d^2*e^2 - (13*B*a^3*b^2 + 18*A*a^2*b^3)*d*e^3)*x
+ 12*(B*a^4*b*d*e^3 - A*a^4*b*e^4 + (B*b^5*d*e^3 - A*b^5*e^4)*x^4 + 4*(B*a*b^4*d
*e^3 - A*a*b^4*e^4)*x^3 + 6*(B*a^2*b^3*d*e^3 - A*a^2*b^3*e^4)*x^2 + 4*(B*a^3*b^2
*d*e^3 - A*a^3*b^2*e^4)*x)*log(b*x + a) - 12*(B*a^4*b*d*e^3 - A*a^4*b*e^4 + (B*b
^5*d*e^3 - A*b^5*e^4)*x^4 + 4*(B*a*b^4*d*e^3 - A*a*b^4*e^4)*x^3 + 6*(B*a^2*b^3*d
*e^3 - A*a^2*b^3*e^4)*x^2 + 4*(B*a^3*b^2*d*e^3 - A*a^3*b^2*e^4)*x)*log(e*x + d))
/(a^4*b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^
8*b^2*d*e^4 - a^9*b*e^5 + (b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^
3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b^5*e^5)*x^4 + 4*(a*b^9*d^5 - 5*a^2*b^8*d^
4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a^6*b^4*e^5)*x
^3 + 6*(a^2*b^8*d^5 - 5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3
+ 5*a^6*b^4*d*e^4 - a^7*b^3*e^5)*x^2 + 4*(a^3*b^7*d^5 - 5*a^4*b^6*d^4*e + 10*a^5
*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5*a^7*b^3*d*e^4 - a^8*b^2*e^5)*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)/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.683781, 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)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

sage0*x