3.1674 \(\int \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=208 \[ -\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^4 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{5 e^4 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^4 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^4 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.204553, antiderivative size = 208, 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 \[ -\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^4 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{5 e^4 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^4 (a+b x)}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.2574, size = 177, normalized size = 0.85 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9 e} + \frac{4 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{2}} + \frac{16 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105 e^{3}} + \frac{32 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 e^{4} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)*(e*x+d)**(1/2),x)

[Out]

2*(d + e*x)**(3/2)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(9*e) + 4*(3*a + 3*b*x)*(
d + e*x)**(3/2)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*e**2) + 16*(d +
e*x)**(3/2)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(105*e**3) + 32*(d
+ e*x)**(3/2)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(315*e**4*(a + b*x
))

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Mathematica [A]  time = 0.137294, size = 120, normalized size = 0.58 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (105 a^3 e^3+63 a^2 b e^2 (3 e x-2 d)+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(105*a^3*e^3 + 63*a^2*b*e^2*(-2*d + 3*e*x)
+ 9*a*b^2*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^3*(-16*d^3 + 24*d^2*e*x - 30*d*e
^2*x^2 + 35*e^3*x^3)))/(315*e^4*(a + b*x))

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Maple [A]  time = 0.01, size = 132, normalized size = 0.6 \[{\frac{70\,{x}^{3}{b}^{3}{e}^{3}+270\,{x}^{2}a{b}^{2}{e}^{3}-60\,{x}^{2}{b}^{3}d{e}^{2}+378\,x{a}^{2}b{e}^{3}-216\,xa{b}^{2}d{e}^{2}+48\,x{b}^{3}{d}^{2}e+210\,{a}^{3}{e}^{3}-252\,{a}^{2}bd{e}^{2}+144\,a{b}^{2}{d}^{2}e-32\,{b}^{3}{d}^{3}}{315\,{e}^{4} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(3/2)*(e*x+d)^(1/2),x)

[Out]

2/315*(e*x+d)^(3/2)*(35*b^3*e^3*x^3+135*a*b^2*e^3*x^2-30*b^3*d*e^2*x^2+189*a^2*b
*e^3*x-108*a*b^2*d*e^2*x+24*b^3*d^2*e*x+105*a^3*e^3-126*a^2*b*d*e^2+72*a*b^2*d^2
*e-16*b^3*d^3)*((b*x+a)^2)^(3/2)/e^4/(b*x+a)^3

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Maxima [A]  time = 0.750668, size = 221, normalized size = 1.06 \[ \frac{2 \,{\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \,{\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} +{\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/315*(35*b^3*e^4*x^4 - 16*b^3*d^4 + 72*a*b^2*d^3*e - 126*a^2*b*d^2*e^2 + 105*a^
3*d*e^3 + 5*(b^3*d*e^3 + 27*a*b^2*e^4)*x^3 - 3*(2*b^3*d^2*e^2 - 9*a*b^2*d*e^3 -
63*a^2*b*e^4)*x^2 + (8*b^3*d^3*e - 36*a*b^2*d^2*e^2 + 63*a^2*b*d*e^3 + 105*a^3*e
^4)*x)*sqrt(e*x + d)/e^4

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Fricas [A]  time = 0.209529, size = 221, normalized size = 1.06 \[ \frac{2 \,{\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \,{\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} +{\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/315*(35*b^3*e^4*x^4 - 16*b^3*d^4 + 72*a*b^2*d^3*e - 126*a^2*b*d^2*e^2 + 105*a^
3*d*e^3 + 5*(b^3*d*e^3 + 27*a*b^2*e^4)*x^3 - 3*(2*b^3*d^2*e^2 - 9*a*b^2*d*e^3 -
63*a^2*b*e^4)*x^2 + (8*b^3*d^3*e - 36*a*b^2*d^2*e^2 + 63*a^2*b*d*e^3 + 105*a^3*e
^4)*x)*sqrt(e*x + d)/e^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)*(e*x+d)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217611, size = 247, normalized size = 1.19 \[ \frac{2}{315} \,{\left (63 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2} b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 9 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a b^{2} e^{\left (-14\right )}{\rm sign}\left (b x + a\right ) +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} b^{3} e^{\left (-27\right )}{\rm sign}\left (b x + a\right ) + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*sqrt(e*x + d),x, algorithm="giac")

[Out]

2/315*(63*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b*e^(-1)*sign(b*x + a) +
9*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2
*e^12)*a*b^2*e^(-14)*sign(b*x + a) + (35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7
/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*b^3*e^
(-27)*sign(b*x + a) + 105*(x*e + d)^(3/2)*a^3*sign(b*x + a))*e^(-1)