3.1745 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=340 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e) \log (d+e x)}{e^7 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (B d-A e)}{e^6 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3}-\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{5 e^2}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b e} \]

[Out]

-((b*(b*d - a*e)^4*(B*d - A*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)))
+ ((b*d - a*e)^3*(B*d - A*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5) -
((b*d - a*e)^2*(B*d - A*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^4) +
((b*d - a*e)*(B*d - A*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3) - (
(B*d - A*e)*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^2) + (B*(a + b*x)^5*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b*e) + ((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*(a + b*x))

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Rubi [A]  time = 0.600639, antiderivative size = 340, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e) \log (d+e x)}{e^7 (a+b x)}-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (B d-A e)}{e^6 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{2 e^5}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3}-\frac{(a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2} (B d-A e)}{5 e^2}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b e} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(b*d - a*e)^4*(B*d - A*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)))
+ ((b*d - a*e)^3*(B*d - A*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^5) -
((b*d - a*e)^2*(B*d - A*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^4) +
((b*d - a*e)*(B*d - A*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3) - (
(B*d - A*e)*(a + b*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^2) + (B*(a + b*x)^5*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b*e) + ((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*(a + b*x))

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Rubi in Sympy [A]  time = 62.1675, size = 294, normalized size = 0.86 \[ \frac{B \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 b e} + \frac{\left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 e^{2}} + \frac{\left (5 a + 5 b x\right ) \left (A e - B d\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{20 e^{3}} + \frac{\left (A e - B d\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{4}} + \frac{\left (3 a + 3 b x\right ) \left (A e - B d\right ) \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e^{5}} + \frac{\left (A e - B d\right ) \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{6}} + \frac{\left (A e - B d\right ) \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{7} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(12*b*e) + (A*e - B*d)*(a**2
+ 2*a*b*x + b**2*x**2)**(5/2)/(5*e**2) + (5*a + 5*b*x)*(A*e - B*d)*(a*e - b*d)*
(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(20*e**3) + (A*e - B*d)*(a*e - b*d)**2*(a**2
+ 2*a*b*x + b**2*x**2)**(3/2)/(3*e**4) + (3*a + 3*b*x)*(A*e - B*d)*(a*e - b*d)*
*3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(6*e**5) + (A*e - B*d)*(a*e - b*d)**4*sqrt(a
**2 + 2*a*b*x + b**2*x**2)/e**6 + (A*e - B*d)*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*x
+ b**2*x**2)*log(d + e*x)/(e**7*(a + b*x))

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Mathematica [A]  time = 0.487544, size = 386, normalized size = 1.14 \[ \frac{\sqrt{(a+b x)^2} \left (e x \left (60 a^5 B e^5+150 a^4 b e^4 (2 A e-2 B d+B e x)+100 a^3 b^2 e^3 \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+50 a^2 b^3 e^2 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+5 a b^4 e \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+b^5 \left (A e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+B \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )\right )+60 (b d-a e)^5 (B d-A e) \log (d+e x)\right )}{60 e^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(e*x*(60*a^5*B*e^5 + 150*a^4*b*e^4*(-2*B*d + 2*A*e + B*e*x) +
100*a^3*b^2*e^3*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 50*a^2
*b^3*e^2*(2*A*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6*d^2*e*x - 4*d*e^2
*x^2 + 3*e^3*x^3)) + 5*a*b^4*e*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3
*x^3) + B*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) +
b^5*(A*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*
(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*
x^5))) + 60*(b*d - a*e)^5*(B*d - A*e)*Log[d + e*x]))/(60*e^7*(a + b*x))

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Maple [B]  time = 0.022, size = 754, normalized size = 2.2 \[{\frac{-300\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+200\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}-15\,A{x}^{4}{b}^{5}d{e}^{5}+150\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+60\,B{x}^{5}a{b}^{4}{e}^{6}+30\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+300\,Ax{a}^{4}b{e}^{6}+60\,Ax{b}^{5}{d}^{4}{e}^{2}+300\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-300\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+150\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-100\,A{x}^{3}a{b}^{4}d{e}^{5}-200\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+100\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-75\,B{x}^{4}a{b}^{4}d{e}^{5}-600\,B\ln \left ( ex+d \right ){a}^{3}{b}^{2}{d}^{3}{e}^{3}+600\,B\ln \left ( ex+d \right ){a}^{2}{b}^{3}{d}^{4}{e}^{2}-300\,B\ln \left ( ex+d \right ) a{b}^{4}{d}^{5}e-300\,A\ln \left ( ex+d \right ){a}^{4}bd{e}^{5}+600\,A\ln \left ( ex+d \right ){a}^{3}{b}^{2}{d}^{2}{e}^{4}-20\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+300\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-30\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+150\,B{x}^{2}{a}^{4}b{e}^{6}+15\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+200\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+20\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}-600\,A\ln \left ( ex+d \right ){a}^{2}{b}^{3}{d}^{3}{e}^{3}+300\,A\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}{e}^{2}+300\,B\ln \left ( ex+d \right ){a}^{4}b{d}^{2}{e}^{4}+600\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-300\,Axa{b}^{4}{d}^{3}{e}^{3}-300\,Bx{a}^{4}bd{e}^{5}+600\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}-600\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+300\,Bxa{b}^{4}{d}^{4}{e}^{2}-150\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}-600\,Ax{a}^{3}{b}^{2}d{e}^{5}+10\,B{x}^{6}{b}^{5}{e}^{6}+12\,A{x}^{5}{b}^{5}{e}^{6}+60\,Bx{a}^{5}{e}^{6}+60\,A\ln \left ( ex+d \right ){a}^{5}{e}^{6}+60\,B\ln \left ( ex+d \right ){b}^{5}{d}^{6}-60\,Bx{b}^{5}{d}^{5}e-60\,A\ln \left ( ex+d \right ){b}^{5}{d}^{5}e-60\,B\ln \left ( ex+d \right ){a}^{5}d{e}^{5}-12\,B{x}^{5}{b}^{5}d{e}^{5}+75\,A{x}^{4}a{b}^{4}{e}^{6}}{60\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*((b*x+a)^2)^(5/2)*(-300*B*x^2*a^3*b^2*d*e^5+200*B*x^3*a^3*b^2*e^6-15*A*x^4*
b^5*d*e^5+150*B*x^4*a^2*b^3*e^6+60*B*x^5*a*b^4*e^6+30*B*x^2*b^5*d^4*e^2+300*A*x*
a^4*b*e^6+60*A*x*b^5*d^4*e^2+300*B*x^2*a^2*b^3*d^2*e^4-300*A*x^2*a^2*b^3*d*e^5+1
50*A*x^2*a*b^4*d^2*e^4-100*A*x^3*a*b^4*d*e^5-200*B*x^3*a^2*b^3*d*e^5+100*B*x^3*a
*b^4*d^2*e^4-75*B*x^4*a*b^4*d*e^5-600*B*ln(e*x+d)*a^3*b^2*d^3*e^3+600*B*ln(e*x+d
)*a^2*b^3*d^4*e^2-300*B*ln(e*x+d)*a*b^4*d^5*e-300*A*ln(e*x+d)*a^4*b*d*e^5+600*A*
ln(e*x+d)*a^3*b^2*d^2*e^4-20*B*x^3*b^5*d^3*e^3+300*A*x^2*a^3*b^2*e^6-30*A*x^2*b^
5*d^3*e^3+150*B*x^2*a^4*b*e^6+15*B*x^4*b^5*d^2*e^4+200*A*x^3*a^2*b^3*e^6+20*A*x^
3*b^5*d^2*e^4-600*A*ln(e*x+d)*a^2*b^3*d^3*e^3+300*A*ln(e*x+d)*a*b^4*d^4*e^2+300*
B*ln(e*x+d)*a^4*b*d^2*e^4+600*A*x*a^2*b^3*d^2*e^4-300*A*x*a*b^4*d^3*e^3-300*B*x*
a^4*b*d*e^5+600*B*x*a^3*b^2*d^2*e^4-600*B*x*a^2*b^3*d^3*e^3+300*B*x*a*b^4*d^4*e^
2-150*B*x^2*a*b^4*d^3*e^3-600*A*x*a^3*b^2*d*e^5+10*B*x^6*b^5*e^6+12*A*x^5*b^5*e^
6+60*B*x*a^5*e^6+60*A*ln(e*x+d)*a^5*e^6+60*B*ln(e*x+d)*b^5*d^6-60*B*x*b^5*d^5*e-
60*A*ln(e*x+d)*b^5*d^5*e-60*B*ln(e*x+d)*a^5*d*e^5-12*B*x^5*b^5*d*e^5+75*A*x^4*a*
b^4*e^6)/(b*x+a)^5/e^7

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280829, size = 749, normalized size = 2.2 \[ \frac{10 \, B b^{5} e^{6} x^{6} - 12 \,{\left (B b^{5} d e^{5} -{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 15 \,{\left (B b^{5} d^{2} e^{4} -{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 20 \,{\left (B b^{5} d^{3} e^{3} -{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 30 \,{\left (B b^{5} d^{4} e^{2} -{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 60 \,{\left (B b^{5} d^{5} e -{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} -{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x + 60 \,{\left (B b^{5} d^{6} + A a^{5} e^{6} -{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} -{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*B*b^5*e^6*x^6 - 12*(B*b^5*d*e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 15*(B*
b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5 + 5*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 2
0*(B*b^5*d^3*e^3 - (5*B*a*b^4 + A*b^5)*d^2*e^4 + 5*(2*B*a^2*b^3 + A*a*b^4)*d*e^5
- 10*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 30*(B*b^5*d^4*e^2 - (5*B*a*b^4 + A*b^5)
*d^3*e^3 + 5*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 10*(B*a^3*b^2 + A*a^2*b^3)*d*e^5
+ 5*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - 60*(B*b^5*d^5*e - (5*B*a*b^4 + A*b^5)*d^4
*e^2 + 5*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 10*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 +
5*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - (B*a^5 + 5*A*a^4*b)*e^6)*x + 60*(B*b^5*d^6 + A
*a^5*e^6 - (5*B*a*b^4 + A*b^5)*d^5*e + 5*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 - 10*(B
*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - (B*a^5 + 5*A
*a^4*b)*d*e^5)*log(e*x + d))/e^7

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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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.304271, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done