3.1689 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=314 \[ \frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^{5/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^6 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) \sqrt{d+e x}}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^6 (a+b x) (d+e x)^{3/2}} \]

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^(9/2)
) - (10*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d + e*x
)^(7/2)) + (4*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d
+ e*x)^(5/2)) - (20*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a
+ b*x)*(d + e*x)^(3/2)) + (10*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^
6*(a + b*x)*Sqrt[d + e*x]) + (2*b^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(e^6*(a + b*x))

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Rubi [A]  time = 0.295498, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^{5/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^6 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) \sqrt{d+e x}}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^6 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^(9/2)
) - (10*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d + e*x
)^(7/2)) + (4*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d
+ e*x)^(5/2)) - (20*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a
+ b*x)*(d + e*x)^(3/2)) + (10*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^
6*(a + b*x)*Sqrt[d + e*x]) + (2*b^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(e^6*(a + b*x))

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Rubi in Sympy [A]  time = 35.7684, size = 255, normalized size = 0.81 \[ \frac{256 b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{5} \sqrt{d + e x}} - \frac{512 b^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{6} \left (a + b x\right ) \sqrt{d + e x}} - \frac{64 b^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{189 e^{4} \left (d + e x\right )^{\frac{3}{2}}} - \frac{32 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{63 e^{3} \left (d + e x\right )^{\frac{5}{2}}} - \frac{4 b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{63 e^{2} \left (d + e x\right )^{\frac{7}{2}}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{9 e \left (d + e x\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(11/2),x)

[Out]

256*b**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*e**5*sqrt(d + e*x)) - 512*b**4*(a*
e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*e**6*(a + b*x)*sqrt(d + e*x)) - 64
*b**3*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(189*e**4*(d + e*x)**(3/2))
- 32*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(63*e**3*(d + e*x)**(5/2)) - 4*b*
(5*a + 5*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(63*e**2*(d + e*x)**(7/2)) - 2
*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(9*e*(d + e*x)**(9/2))

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Mathematica [A]  time = 0.361519, size = 143, normalized size = 0.46 \[ \frac{2 \left ((a+b x)^2\right )^{5/2} \sqrt{d+e x} \left (\frac{315 b^4 (b d-a e)}{d+e x}-\frac{210 b^3 (b d-a e)^2}{(d+e x)^2}+\frac{126 b^2 (b d-a e)^3}{(d+e x)^3}-\frac{45 b (b d-a e)^4}{(d+e x)^4}+\frac{7 (b d-a e)^5}{(d+e x)^5}+63 b^5\right )}{63 e^6 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(2*((a + b*x)^2)^(5/2)*Sqrt[d + e*x]*(63*b^5 + (7*(b*d - a*e)^5)/(d + e*x)^5 - (
45*b*(b*d - a*e)^4)/(d + e*x)^4 + (126*b^2*(b*d - a*e)^3)/(d + e*x)^3 - (210*b^3
*(b*d - a*e)^2)/(d + e*x)^2 + (315*b^4*(b*d - a*e))/(d + e*x)))/(63*e^6*(a + b*x
)^5)

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Maple [A]  time = 0.01, size = 289, normalized size = 0.9 \[ -{\frac{-126\,{x}^{5}{b}^{5}{e}^{5}+630\,{x}^{4}a{b}^{4}{e}^{5}-1260\,{x}^{4}{b}^{5}d{e}^{4}+420\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+1680\,{x}^{3}a{b}^{4}d{e}^{4}-3360\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+252\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+504\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+2016\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-4032\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+90\,x{a}^{4}b{e}^{5}+144\,x{a}^{3}{b}^{2}d{e}^{4}+288\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+1152\,xa{b}^{4}{d}^{3}{e}^{2}-2304\,x{b}^{5}{d}^{4}e+14\,{a}^{5}{e}^{5}+20\,{a}^{4}bd{e}^{4}+32\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+64\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+256\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{63\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x)

[Out]

-2/63/(e*x+d)^(9/2)*(-63*b^5*e^5*x^5+315*a*b^4*e^5*x^4-630*b^5*d*e^4*x^4+210*a^2
*b^3*e^5*x^3+840*a*b^4*d*e^4*x^3-1680*b^5*d^2*e^3*x^3+126*a^3*b^2*e^5*x^2+252*a^
2*b^3*d*e^4*x^2+1008*a*b^4*d^2*e^3*x^2-2016*b^5*d^3*e^2*x^2+45*a^4*b*e^5*x+72*a^
3*b^2*d*e^4*x+144*a^2*b^3*d^2*e^3*x+576*a*b^4*d^3*e^2*x-1152*b^5*d^4*e*x+7*a^5*e
^5+10*a^4*b*d*e^4+16*a^3*b^2*d^2*e^3+32*a^2*b^3*d^3*e^2+128*a*b^4*d^4*e-256*b^5*
d^5)*((b*x+a)^2)^(5/2)/e^6/(b*x+a)^5

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Maxima [A]  time = 0.738423, size = 412, normalized size = 1.31 \[ \frac{2 \,{\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \,{\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \,{\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \,{\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \,{\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )}}{63 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

2/63*(63*b^5*e^5*x^5 + 256*b^5*d^5 - 128*a*b^4*d^4*e - 32*a^2*b^3*d^3*e^2 - 16*a
^3*b^2*d^2*e^3 - 10*a^4*b*d*e^4 - 7*a^5*e^5 + 315*(2*b^5*d*e^4 - a*b^4*e^5)*x^4
+ 210*(8*b^5*d^2*e^3 - 4*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 126*(16*b^5*d^3*e^2 -
8*a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 9*(128*b^5*d^4*e - 64*a*b
^4*d^3*e^2 - 16*a^2*b^3*d^2*e^3 - 8*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)/((e^10*x^4 +
4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)*sqrt(e*x + d))

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Fricas [A]  time = 0.208951, size = 412, normalized size = 1.31 \[ \frac{2 \,{\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \,{\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \,{\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \,{\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \,{\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )}}{63 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

2/63*(63*b^5*e^5*x^5 + 256*b^5*d^5 - 128*a*b^4*d^4*e - 32*a^2*b^3*d^3*e^2 - 16*a
^3*b^2*d^2*e^3 - 10*a^4*b*d*e^4 - 7*a^5*e^5 + 315*(2*b^5*d*e^4 - a*b^4*e^5)*x^4
+ 210*(8*b^5*d^2*e^3 - 4*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 126*(16*b^5*d^3*e^2 -
8*a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 9*(128*b^5*d^4*e - 64*a*b
^4*d^3*e^2 - 16*a^2*b^3*d^2*e^3 - 8*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)/((e^10*x^4 +
4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)*sqrt(e*x + d))

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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**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234115, size = 608, normalized size = 1.94 \[ 2 \, \sqrt{x e + d} b^{5} e^{\left (-6\right )}{\rm sign}\left (b x + a\right ) + \frac{2 \,{\left (315 \,{\left (x e + d\right )}^{4} b^{5} d{\rm sign}\left (b x + a\right ) - 210 \,{\left (x e + d\right )}^{3} b^{5} d^{2}{\rm sign}\left (b x + a\right ) + 126 \,{\left (x e + d\right )}^{2} b^{5} d^{3}{\rm sign}\left (b x + a\right ) - 45 \,{\left (x e + d\right )} b^{5} d^{4}{\rm sign}\left (b x + a\right ) + 7 \, b^{5} d^{5}{\rm sign}\left (b x + a\right ) - 315 \,{\left (x e + d\right )}^{4} a b^{4} e{\rm sign}\left (b x + a\right ) + 420 \,{\left (x e + d\right )}^{3} a b^{4} d e{\rm sign}\left (b x + a\right ) - 378 \,{\left (x e + d\right )}^{2} a b^{4} d^{2} e{\rm sign}\left (b x + a\right ) + 180 \,{\left (x e + d\right )} a b^{4} d^{3} e{\rm sign}\left (b x + a\right ) - 35 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) - 210 \,{\left (x e + d\right )}^{3} a^{2} b^{3} e^{2}{\rm sign}\left (b x + a\right ) + 378 \,{\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2}{\rm sign}\left (b x + a\right ) - 270 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 70 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) - 126 \,{\left (x e + d\right )}^{2} a^{3} b^{2} e^{3}{\rm sign}\left (b x + a\right ) + 180 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3}{\rm sign}\left (b x + a\right ) - 70 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) - 45 \,{\left (x e + d\right )} a^{4} b e^{4}{\rm sign}\left (b x + a\right ) + 35 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) - 7 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{63 \,{\left (x e + d\right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*b^5*e^(-6)*sign(b*x + a) + 2/63*(315*(x*e + d)^4*b^5*d*sign(b*x
+ a) - 210*(x*e + d)^3*b^5*d^2*sign(b*x + a) + 126*(x*e + d)^2*b^5*d^3*sign(b*x
+ a) - 45*(x*e + d)*b^5*d^4*sign(b*x + a) + 7*b^5*d^5*sign(b*x + a) - 315*(x*e +
d)^4*a*b^4*e*sign(b*x + a) + 420*(x*e + d)^3*a*b^4*d*e*sign(b*x + a) - 378*(x*e
+ d)^2*a*b^4*d^2*e*sign(b*x + a) + 180*(x*e + d)*a*b^4*d^3*e*sign(b*x + a) - 35
*a*b^4*d^4*e*sign(b*x + a) - 210*(x*e + d)^3*a^2*b^3*e^2*sign(b*x + a) + 378*(x*
e + d)^2*a^2*b^3*d*e^2*sign(b*x + a) - 270*(x*e + d)*a^2*b^3*d^2*e^2*sign(b*x +
a) + 70*a^2*b^3*d^3*e^2*sign(b*x + a) - 126*(x*e + d)^2*a^3*b^2*e^3*sign(b*x + a
) + 180*(x*e + d)*a^3*b^2*d*e^3*sign(b*x + a) - 70*a^3*b^2*d^2*e^3*sign(b*x + a)
- 45*(x*e + d)*a^4*b*e^4*sign(b*x + a) + 35*a^4*b*d*e^4*sign(b*x + a) - 7*a^5*e
^5*sign(b*x + a))*e^(-6)/(x*e + d)^(9/2)