3.1850 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=304 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{e^5 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^5 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.468858, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{e^5 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 55.5699, size = 320, normalized size = 1.05 \[ \frac{16 b^{2} \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e + 5 B a e - 8 B b d\right )}{15 e^{4} \left (a e - b d\right )} + \frac{32 b^{2} \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e + 5 B a e - 8 B b d\right )}{15 e^{5} \left (a + b x\right )} - \frac{4 b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e + 5 B a e - 8 B b d\right )}{15 e^{3} \sqrt{d + e x} \left (a e - b d\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (3 A b e + 5 B a e - 8 B b d\right )}{15 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

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)**(3/2)/(e*x+d)**(7/2),x)

[Out]

16*b**2*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(3*A*b*e + 5*B*a*e - 8*B*
b*d)/(15*e**4*(a*e - b*d)) + 32*b**2*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x*
*2)*(3*A*b*e + 5*B*a*e - 8*B*b*d)/(15*e**5*(a + b*x)) - 4*b*(3*a + 3*b*x)*sqrt(a
**2 + 2*a*b*x + b**2*x**2)*(3*A*b*e + 5*B*a*e - 8*B*b*d)/(15*e**3*sqrt(d + e*x)*
(a*e - b*d)) - (2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(5*
e*(d + e*x)**(5/2)*(a*e - b*d)) - 2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(3*A*b*e
+ 5*B*a*e - 8*B*b*d)/(15*e**2*(d + e*x)**(3/2)*(a*e - b*d))

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Mathematica [A]  time = 0.409851, size = 157, normalized size = 0.52 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} \sqrt{d+e x} \left (-5 b^2 (-9 a B e-3 A b e+11 b B d)+\frac{45 b (b d-a e) (a B e+A b e-2 b B d)}{d+e x}-\frac{5 (b d-a e)^2 (a B e+3 A b e-4 b B d)}{(d+e x)^2}-\frac{3 (b d-a e)^3 (B d-A e)}{(d+e x)^3}+5 b^3 B e x\right )}{15 e^5 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

(2*((a + b*x)^2)^(3/2)*Sqrt[d + e*x]*(-5*b^2*(11*b*B*d - 3*A*b*e - 9*a*B*e) + 5*
b^3*B*e*x - (3*(b*d - a*e)^3*(B*d - A*e))/(d + e*x)^3 - (5*(b*d - a*e)^2*(-4*b*B
*d + 3*A*b*e + a*B*e))/(d + e*x)^2 + (45*b*(b*d - a*e)*(-2*b*B*d + A*b*e + a*B*e
))/(d + e*x)))/(15*e^5*(a + b*x)^3)

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Maple [A]  time = 0.014, size = 317, normalized size = 1. \[ -{\frac{-10\,B{x}^{4}{b}^{3}{e}^{4}-30\,A{x}^{3}{b}^{3}{e}^{4}-90\,B{x}^{3}a{b}^{2}{e}^{4}+80\,B{x}^{3}{b}^{3}d{e}^{3}+90\,A{x}^{2}a{b}^{2}{e}^{4}-180\,A{x}^{2}{b}^{3}d{e}^{3}+90\,B{x}^{2}{a}^{2}b{e}^{4}-540\,B{x}^{2}a{b}^{2}d{e}^{3}+480\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+30\,Ax{a}^{2}b{e}^{4}+120\,Axa{b}^{2}d{e}^{3}-240\,Ax{b}^{3}{d}^{2}{e}^{2}+10\,Bx{a}^{3}{e}^{4}+120\,Bx{a}^{2}bd{e}^{3}-720\,Bxa{b}^{2}{d}^{2}{e}^{2}+640\,Bx{b}^{3}{d}^{3}e+6\,A{a}^{3}{e}^{4}+12\,Ad{e}^{3}{a}^{2}b+48\,Aa{b}^{2}{d}^{2}{e}^{2}-96\,A{b}^{3}{d}^{3}e+4\,Bd{e}^{3}{a}^{3}+48\,B{a}^{2}b{d}^{2}{e}^{2}-288\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{15\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

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)^(3/2)/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(-5*B*b^3*e^4*x^4-15*A*b^3*e^4*x^3-45*B*a*b^2*e^4*x^3+40*B*b
^3*d*e^3*x^3+45*A*a*b^2*e^4*x^2-90*A*b^3*d*e^3*x^2+45*B*a^2*b*e^4*x^2-270*B*a*b^
2*d*e^3*x^2+240*B*b^3*d^2*e^2*x^2+15*A*a^2*b*e^4*x+60*A*a*b^2*d*e^3*x-120*A*b^3*
d^2*e^2*x+5*B*a^3*e^4*x+60*B*a^2*b*d*e^3*x-360*B*a*b^2*d^2*e^2*x+320*B*b^3*d^3*e
*x+3*A*a^3*e^4+6*A*a^2*b*d*e^3+24*A*a*b^2*d^2*e^2-48*A*b^3*d^3*e+2*B*a^3*d*e^3+2
4*B*a^2*b*d^2*e^2-144*B*a*b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)
^3

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Maxima [A]  time = 0.749571, size = 440, normalized size = 1.45 \[ \frac{2 \,{\left (5 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} - a^{3} e^{3} + 15 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (8 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} A}{5 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (5 \, b^{3} e^{4} x^{4} - 128 \, b^{3} d^{4} + 144 \, a b^{2} d^{3} e - 24 \, a^{2} b d^{2} e^{2} - 2 \, a^{3} d e^{3} - 5 \,{\left (8 \, b^{3} d e^{3} - 9 \, a b^{2} e^{4}\right )} x^{3} - 15 \,{\left (16 \, b^{3} d^{2} e^{2} - 18 \, a b^{2} d e^{3} + 3 \, a^{2} b e^{4}\right )} x^{2} - 5 \,{\left (64 \, b^{3} d^{3} e - 72 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} B}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/5*(5*b^3*e^3*x^3 + 16*b^3*d^3 - 8*a*b^2*d^2*e - 2*a^2*b*d*e^2 - a^3*e^3 + 15*(
2*b^3*d*e^2 - a*b^2*e^3)*x^2 + 5*(8*b^3*d^2*e - 4*a*b^2*d*e^2 - a^2*b*e^3)*x)*A/
((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(e*x + d)) + 2/15*(5*b^3*e^4*x^4 - 128*b^3*
d^4 + 144*a*b^2*d^3*e - 24*a^2*b*d^2*e^2 - 2*a^3*d*e^3 - 5*(8*b^3*d*e^3 - 9*a*b^
2*e^4)*x^3 - 15*(16*b^3*d^2*e^2 - 18*a*b^2*d*e^3 + 3*a^2*b*e^4)*x^2 - 5*(64*b^3*
d^3*e - 72*a*b^2*d^2*e^2 + 12*a^2*b*d*e^3 + a^3*e^4)*x)*B/((e^7*x^2 + 2*d*e^6*x
+ d^2*e^5)*sqrt(e*x + d))

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Fricas [A]  time = 0.298421, size = 382, normalized size = 1.26 \[ \frac{2 \,{\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} + 48 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 24 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \,{\left (8 \, B b^{3} d e^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} - 15 \,{\left (16 \, B b^{3} d^{2} e^{2} - 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 5 \,{\left (64 \, B b^{3} d^{3} e - 24 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*B*b^3*e^4*x^4 - 128*B*b^3*d^4 - 3*A*a^3*e^4 + 48*(3*B*a*b^2 + A*b^3)*d^3
*e - 24*(B*a^2*b + A*a*b^2)*d^2*e^2 - 2*(B*a^3 + 3*A*a^2*b)*d*e^3 - 5*(8*B*b^3*d
*e^3 - 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 - 15*(16*B*b^3*d^2*e^2 - 6*(3*B*a*b^2 + A*
b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 - 5*(64*B*b^3*d^3*e - 24*(3*B*a*b^2
+ A*b^3)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x)/((
e^7*x^2 + 2*d*e^6*x + d^2*e^5)*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*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.307057, size = 686, normalized size = 2.26 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{3} e^{10}{\rm sign}\left (b x + a\right ) - 12 \, \sqrt{x e + d} B b^{3} d e^{10}{\rm sign}\left (b x + a\right ) + 9 \, \sqrt{x e + d} B a b^{2} e^{11}{\rm sign}\left (b x + a\right ) + 3 \, \sqrt{x e + d} A b^{3} e^{11}{\rm sign}\left (b x + a\right )\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} B b^{3} d^{2}{\rm sign}\left (b x + a\right ) - 20 \,{\left (x e + d\right )} B b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 3 \, B b^{3} d^{4}{\rm sign}\left (b x + a\right ) - 135 \,{\left (x e + d\right )}^{2} B a b^{2} d e{\rm sign}\left (b x + a\right ) - 45 \,{\left (x e + d\right )}^{2} A b^{3} d e{\rm sign}\left (b x + a\right ) + 45 \,{\left (x e + d\right )} B a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 15 \,{\left (x e + d\right )} A b^{3} d^{2} e{\rm sign}\left (b x + a\right ) - 9 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) - 3 \, A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 45 \,{\left (x e + d\right )}^{2} B a^{2} b e^{2}{\rm sign}\left (b x + a\right ) + 45 \,{\left (x e + d\right )}^{2} A a b^{2} e^{2}{\rm sign}\left (b x + a\right ) - 30 \,{\left (x e + d\right )} B a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) - 30 \,{\left (x e + d\right )} A a b^{2} d e^{2}{\rm sign}\left (b x + a\right ) + 9 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 9 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )} B a^{3} e^{3}{\rm sign}\left (b x + a\right ) + 15 \,{\left (x e + d\right )} A a^{2} b e^{3}{\rm sign}\left (b x + a\right ) - 3 \, B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) - 9 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 3 \, A a^{3} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*b^3*e^10*sign(b*x + a) - 12*sqrt(x*e + d)*B*b^3*d*e^10*si
gn(b*x + a) + 9*sqrt(x*e + d)*B*a*b^2*e^11*sign(b*x + a) + 3*sqrt(x*e + d)*A*b^3
*e^11*sign(b*x + a))*e^(-15) - 2/15*(90*(x*e + d)^2*B*b^3*d^2*sign(b*x + a) - 20
*(x*e + d)*B*b^3*d^3*sign(b*x + a) + 3*B*b^3*d^4*sign(b*x + a) - 135*(x*e + d)^2
*B*a*b^2*d*e*sign(b*x + a) - 45*(x*e + d)^2*A*b^3*d*e*sign(b*x + a) + 45*(x*e +
d)*B*a*b^2*d^2*e*sign(b*x + a) + 15*(x*e + d)*A*b^3*d^2*e*sign(b*x + a) - 9*B*a*
b^2*d^3*e*sign(b*x + a) - 3*A*b^3*d^3*e*sign(b*x + a) + 45*(x*e + d)^2*B*a^2*b*e
^2*sign(b*x + a) + 45*(x*e + d)^2*A*a*b^2*e^2*sign(b*x + a) - 30*(x*e + d)*B*a^2
*b*d*e^2*sign(b*x + a) - 30*(x*e + d)*A*a*b^2*d*e^2*sign(b*x + a) + 9*B*a^2*b*d^
2*e^2*sign(b*x + a) + 9*A*a*b^2*d^2*e^2*sign(b*x + a) + 5*(x*e + d)*B*a^3*e^3*si
gn(b*x + a) + 15*(x*e + d)*A*a^2*b*e^3*sign(b*x + a) - 3*B*a^3*d*e^3*sign(b*x +
a) - 9*A*a^2*b*d*e^3*sign(b*x + a) + 3*A*a^3*e^4*sign(b*x + a))*e^(-5)/(x*e + d)
^(5/2)