3.2092 \(\int (a+b x) (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=152 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^3 (a+b x)} \]

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^3*(a + b*x)
) - (4*b*(b*d - a*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^3*(a
+ b*x)) + (2*b^2*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^3*(a + b*
x))

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Rubi [A]  time = 0.229445, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^3*(a + b*x)
) - (4*b*(b*d - a*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^3*(a
+ b*x)) + (2*b^2*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^3*(a + b*
x))

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Rubi in Sympy [A]  time = 25.255, size = 129, normalized size = 0.85 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{9}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{13 e} + \frac{8 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{143 e^{2}} + \frac{16 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{1287 e^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(7/2)*((b*x+a)**2)**(1/2),x)

[Out]

2*(a + b*x)*(d + e*x)**(9/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(13*e) + 8*(d + e*
x)**(9/2)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(143*e**2) + 16*(d + e*x)
**(9/2)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(1287*e**3*(a + b*x))

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Mathematica [A]  time = 0.14999, size = 79, normalized size = 0.52 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(9/2)*(143*a^2*e^2 + 26*a*b*e*(-2*d + 9*e*x) + b^
2*(8*d^2 - 36*d*e*x + 99*e^2*x^2)))/(1287*e^3*(a + b*x))

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Maple [A]  time = 0.009, size = 79, normalized size = 0.5 \[{\frac{198\,{x}^{2}{b}^{2}{e}^{2}+468\,xab{e}^{2}-72\,x{b}^{2}de+286\,{a}^{2}{e}^{2}-104\,abde+16\,{b}^{2}{d}^{2}}{1287\,{e}^{3} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2),x)

[Out]

2/1287*(e*x+d)^(9/2)*(99*b^2*e^2*x^2+234*a*b*e^2*x-36*b^2*d*e*x+143*a^2*e^2-52*a
*b*d*e+8*b^2*d^2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)

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Maxima [A]  time = 0.735996, size = 355, normalized size = 2.34 \[ \frac{2 \,{\left (9 \, b e^{5} x^{5} - 2 \, b d^{5} + 11 \, a d^{4} e +{\left (34 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + 2 \,{\left (23 \, b d^{2} e^{3} + 22 \, a d e^{4}\right )} x^{3} + 6 \,{\left (4 \, b d^{3} e^{2} + 11 \, a d^{2} e^{3}\right )} x^{2} +{\left (b d^{4} e + 44 \, a d^{3} e^{2}\right )} x\right )} \sqrt{e x + d} a}{99 \, e^{2}} + \frac{2 \,{\left (99 \, b e^{6} x^{6} + 8 \, b d^{6} - 26 \, a d^{5} e + 9 \,{\left (40 \, b d e^{5} + 13 \, a e^{6}\right )} x^{5} + 2 \,{\left (229 \, b d^{2} e^{4} + 221 \, a d e^{5}\right )} x^{4} + 2 \,{\left (106 \, b d^{3} e^{3} + 299 \, a d^{2} e^{4}\right )} x^{3} + 3 \,{\left (b d^{4} e^{2} + 104 \, a d^{3} e^{3}\right )} x^{2} -{\left (4 \, b d^{5} e - 13 \, a d^{4} e^{2}\right )} x\right )} \sqrt{e x + d} b}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/99*(9*b*e^5*x^5 - 2*b*d^5 + 11*a*d^4*e + (34*b*d*e^4 + 11*a*e^5)*x^4 + 2*(23*b
*d^2*e^3 + 22*a*d*e^4)*x^3 + 6*(4*b*d^3*e^2 + 11*a*d^2*e^3)*x^2 + (b*d^4*e + 44*
a*d^3*e^2)*x)*sqrt(e*x + d)*a/e^2 + 2/1287*(99*b*e^6*x^6 + 8*b*d^6 - 26*a*d^5*e
+ 9*(40*b*d*e^5 + 13*a*e^6)*x^5 + 2*(229*b*d^2*e^4 + 221*a*d*e^5)*x^4 + 2*(106*b
*d^3*e^3 + 299*a*d^2*e^4)*x^3 + 3*(b*d^4*e^2 + 104*a*d^3*e^3)*x^2 - (4*b*d^5*e -
13*a*d^4*e^2)*x)*sqrt(e*x + d)*b/e^3

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Fricas [A]  time = 0.276131, size = 286, normalized size = 1.88 \[ \frac{2 \,{\left (99 \, b^{2} e^{6} x^{6} + 8 \, b^{2} d^{6} - 52 \, a b d^{5} e + 143 \, a^{2} d^{4} e^{2} + 18 \,{\left (20 \, b^{2} d e^{5} + 13 \, a b e^{6}\right )} x^{5} +{\left (458 \, b^{2} d^{2} e^{4} + 884 \, a b d e^{5} + 143 \, a^{2} e^{6}\right )} x^{4} + 4 \,{\left (53 \, b^{2} d^{3} e^{3} + 299 \, a b d^{2} e^{4} + 143 \, a^{2} d e^{5}\right )} x^{3} + 3 \,{\left (b^{2} d^{4} e^{2} + 208 \, a b d^{3} e^{3} + 286 \, a^{2} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{2} d^{5} e - 13 \, a b d^{4} e^{2} - 286 \, a^{2} d^{3} e^{3}\right )} x\right )} \sqrt{e x + d}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1287*(99*b^2*e^6*x^6 + 8*b^2*d^6 - 52*a*b*d^5*e + 143*a^2*d^4*e^2 + 18*(20*b^2
*d*e^5 + 13*a*b*e^6)*x^5 + (458*b^2*d^2*e^4 + 884*a*b*d*e^5 + 143*a^2*e^6)*x^4 +
4*(53*b^2*d^3*e^3 + 299*a*b*d^2*e^4 + 143*a^2*d*e^5)*x^3 + 3*(b^2*d^4*e^2 + 208
*a*b*d^3*e^3 + 286*a^2*d^2*e^4)*x^2 - 2*(2*b^2*d^5*e - 13*a*b*d^4*e^2 - 286*a^2*
d^3*e^3)*x)*sqrt(e*x + d)/e^3

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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)*(e*x+d)**(7/2)*((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.31428, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done