3.1685 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.297597, 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 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}{e^6 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x) \sqrt{d+e x}}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^6 (a+b x)}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 40.3292, size = 269, normalized size = 0.86 \[ \frac{20 b \left (a + b x\right ) \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9 e^{2}} + \frac{160 b \sqrt{d + e x} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{63 e^{3}} + \frac{64 b \left (3 a + 3 b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{4}} + \frac{256 b \sqrt{d + e x} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{5}} + \frac{512 b \sqrt{d + e x} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{6} \left (a + b x\right )} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{e \sqrt{d + e x}} \]

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)**(3/2),x)

[Out]

20*b*(a + b*x)*sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(9*e**2) + 160*
b*sqrt(d + e*x)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(63*e**3) + 64*b
*(3*a + 3*b*x)*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63
*e**4) + 256*b*sqrt(d + e*x)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63
*e**5) + 512*b*sqrt(d + e*x)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63
*e**6*(a + b*x)) - 2*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(e*sqrt(d + e*x))

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Mathematica [A]  time = 0.27187, size = 234, normalized size = 0.75 \[ -\frac{2 \sqrt{(a+b x)^2} \left (63 a^5 e^5-315 a^4 b e^4 (2 d+e x)+210 a^3 b^2 e^3 \left (8 d^2+4 d e x-e^2 x^2\right )-126 a^2 b^3 e^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+9 a b^4 e \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )+b^5 \left (-\left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )\right )}{63 e^6 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 289, normalized size = 0.9 \[ -{\frac{-14\,{x}^{5}{b}^{5}{e}^{5}-90\,{x}^{4}a{b}^{4}{e}^{5}+20\,{x}^{4}{b}^{5}d{e}^{4}-252\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+144\,{x}^{3}a{b}^{4}d{e}^{4}-32\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-420\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+504\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-288\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+64\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-630\,x{a}^{4}b{e}^{5}+1680\,x{a}^{3}{b}^{2}d{e}^{4}-2016\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+1152\,xa{b}^{4}{d}^{3}{e}^{2}-256\,x{b}^{5}{d}^{4}e+126\,{a}^{5}{e}^{5}-1260\,{a}^{4}bd{e}^{4}+3360\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-4032\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+2304\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{63\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(3/2),x)

[Out]

-2/63/(e*x+d)^(1/2)*(-7*b^5*e^5*x^5-45*a*b^4*e^5*x^4+10*b^5*d*e^4*x^4-126*a^2*b^
3*e^5*x^3+72*a*b^4*d*e^4*x^3-16*b^5*d^2*e^3*x^3-210*a^3*b^2*e^5*x^2+252*a^2*b^3*
d*e^4*x^2-144*a*b^4*d^2*e^3*x^2+32*b^5*d^3*e^2*x^2-315*a^4*b*e^5*x+840*a^3*b^2*d
*e^4*x-1008*a^2*b^3*d^2*e^3*x+576*a*b^4*d^3*e^2*x-128*b^5*d^4*e*x+63*a^5*e^5-630
*a^4*b*d*e^4+1680*a^3*b^2*d^2*e^3-2016*a^2*b^3*d^3*e^2+1152*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 = 0.736398, size = 352, normalized size = 1.12 \[ \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \,{\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )}}{63 \, \sqrt{e x + d} e^{6}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 16
80*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5
)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*x^3 - 2*(16*b^5*d^3*
e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4
*e - 576*a*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^
5)*x)/(sqrt(e*x + d)*e^6)

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Fricas [A]  time = 0.208667, size = 352, normalized size = 1.12 \[ \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \,{\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )}}{63 \, \sqrt{e x + d} e^{6}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 16
80*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5
)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*x^3 - 2*(16*b^5*d^3*
e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4
*e - 576*a*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^
5)*x)/(sqrt(e*x + d)*e^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral(((a + b*x)**2)**(5/2)/(d + e*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.228738, size = 637, normalized size = 2.03 \[ \frac{2}{63} \,{\left (7 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{5} e^{48}{\rm sign}\left (b x + a\right ) - 45 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d e^{48}{\rm sign}\left (b x + a\right ) + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{2} e^{48}{\rm sign}\left (b x + a\right ) - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{3} e^{48}{\rm sign}\left (b x + a\right ) + 315 \, \sqrt{x e + d} b^{5} d^{4} e^{48}{\rm sign}\left (b x + a\right ) + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} e^{49}{\rm sign}\left (b x + a\right ) - 252 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d e^{49}{\rm sign}\left (b x + a\right ) + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{2} e^{49}{\rm sign}\left (b x + a\right ) - 1260 \, \sqrt{x e + d} a b^{4} d^{3} e^{49}{\rm sign}\left (b x + a\right ) + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} e^{50}{\rm sign}\left (b x + a\right ) - 630 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d e^{50}{\rm sign}\left (b x + a\right ) + 1890 \, \sqrt{x e + d} a^{2} b^{3} d^{2} e^{50}{\rm sign}\left (b x + a\right ) + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} e^{51}{\rm sign}\left (b x + a\right ) - 1260 \, \sqrt{x e + d} a^{3} b^{2} d e^{51}{\rm sign}\left (b x + a\right ) + 315 \, \sqrt{x e + d} a^{4} b e^{52}{\rm sign}\left (b x + a\right )\right )} e^{\left (-54\right )} + \frac{2 \,{\left (b^{5} d^{5}{\rm sign}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) - a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{\sqrt{x e + d}} \]

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)^(3/2),x, algorithm="giac")

[Out]

2/63*(7*(x*e + d)^(9/2)*b^5*e^48*sign(b*x + a) - 45*(x*e + d)^(7/2)*b^5*d*e^48*s
ign(b*x + a) + 126*(x*e + d)^(5/2)*b^5*d^2*e^48*sign(b*x + a) - 210*(x*e + d)^(3
/2)*b^5*d^3*e^48*sign(b*x + a) + 315*sqrt(x*e + d)*b^5*d^4*e^48*sign(b*x + a) +
45*(x*e + d)^(7/2)*a*b^4*e^49*sign(b*x + a) - 252*(x*e + d)^(5/2)*a*b^4*d*e^49*s
ign(b*x + a) + 630*(x*e + d)^(3/2)*a*b^4*d^2*e^49*sign(b*x + a) - 1260*sqrt(x*e
+ d)*a*b^4*d^3*e^49*sign(b*x + a) + 126*(x*e + d)^(5/2)*a^2*b^3*e^50*sign(b*x +
a) - 630*(x*e + d)^(3/2)*a^2*b^3*d*e^50*sign(b*x + a) + 1890*sqrt(x*e + d)*a^2*b
^3*d^2*e^50*sign(b*x + a) + 210*(x*e + d)^(3/2)*a^3*b^2*e^51*sign(b*x + a) - 126
0*sqrt(x*e + d)*a^3*b^2*d*e^51*sign(b*x + a) + 315*sqrt(x*e + d)*a^4*b*e^52*sign
(b*x + a))*e^(-54) + 2*(b^5*d^5*sign(b*x + a) - 5*a*b^4*d^4*e*sign(b*x + a) + 10
*a^2*b^3*d^3*e^2*sign(b*x + a) - 10*a^3*b^2*d^2*e^3*sign(b*x + a) + 5*a^4*b*d*e^
4*sign(b*x + a) - a^5*e^5*sign(b*x + a))*e^(-6)/sqrt(x*e + d)