3.349 \(\int \frac{\left (b x+c x^2\right )^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=145 \[ \frac{2 (d+e x)^{5/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{5 e^5}+\frac{2 d^2 \sqrt{d+e x} (c d-b e)^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}-\frac{4 d (d+e x)^{3/2} (c d-b e) (2 c d-b e)}{3 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5} \]

[Out]

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

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Rubi [A]  time = 0.198471, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 (d+e x)^{5/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{5 e^5}+\frac{2 d^2 \sqrt{d+e x} (c d-b e)^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}-\frac{4 d (d+e x)^{3/2} (c d-b e) (2 c d-b e)}{3 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.9906, size = 139, normalized size = 0.96 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )}{7 e^{5}} + \frac{2 d^{2} \sqrt{d + e x} \left (b e - c d\right )^{2}}{e^{5}} - \frac{4 d \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )}{3 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{5 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**2*(d + e*x)**(9/2)/(9*e**5) + 4*c*(d + e*x)**(7/2)*(b*e - 2*c*d)/(7*e**5) +
2*d**2*sqrt(d + e*x)*(b*e - c*d)**2/e**5 - 4*d*(d + e*x)**(3/2)*(b*e - 2*c*d)*(
b*e - c*d)/(3*e**5) + 2*(d + e*x)**(5/2)*(b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)/(
5*e**5)

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Mathematica [A]  time = 0.103587, size = 124, normalized size = 0.86 \[ \frac{2 \sqrt{d+e x} \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(21*b^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 18*b*c*e*(-16*d^3 +
8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x
^2 - 40*d*e^3*x^3 + 35*e^4*x^4)))/(315*e^5)

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Maple [A]  time = 0.01, size = 141, normalized size = 1. \[{\frac{70\,{c}^{2}{x}^{4}{e}^{4}+180\,bc{e}^{4}{x}^{3}-80\,{c}^{2}d{e}^{3}{x}^{3}+126\,{b}^{2}{e}^{4}{x}^{2}-216\,bcd{e}^{3}{x}^{2}+96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-168\,{b}^{2}d{e}^{3}x+288\,bc{d}^{2}{e}^{2}x-128\,{c}^{2}{d}^{3}ex+336\,{b}^{2}{d}^{2}{e}^{2}-576\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*c^2*e^4*x^4+90*b*c*e^4*x^3-40*c^2*d*e^3*x^3+63*b^2*e^4*x^2-108*b*c*d*e
^3*x^2+48*c^2*d^2*e^2*x^2-84*b^2*d*e^3*x+144*b*c*d^2*e^2*x-64*c^2*d^3*e*x+168*b^
2*d^2*e^2-288*b*c*d^3*e+128*c^2*d^4)*(e*x+d)^(1/2)/e^5

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Maxima [A]  time = 0.697106, size = 216, normalized size = 1.49 \[ \frac{2 \,{\left (\frac{21 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} b^{2}}{e^{2}} + \frac{18 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )} b c}{e^{3}} + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(21*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2/
e^2 + 18*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35
*sqrt(e*x + d)*d^3)*b*c/e^3 + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*
(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2/e^4)/
e

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Fricas [A]  time = 0.212508, size = 186, normalized size = 1.28 \[ \frac{2 \,{\left (35 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 288 \, b c d^{3} e + 168 \, b^{2} d^{2} e^{2} - 10 \,{\left (4 \, c^{2} d e^{3} - 9 \, b c e^{4}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{2} e^{2} - 36 \, b c d e^{3} + 21 \, b^{2} e^{4}\right )} x^{2} - 4 \,{\left (16 \, c^{2} d^{3} e - 36 \, b c d^{2} e^{2} + 21 \, b^{2} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*c^2*e^4*x^4 + 128*c^2*d^4 - 288*b*c*d^3*e + 168*b^2*d^2*e^2 - 10*(4*c^
2*d*e^3 - 9*b*c*e^4)*x^3 + 3*(16*c^2*d^2*e^2 - 36*b*c*d*e^3 + 21*b^2*e^4)*x^2 -
4*(16*c^2*d^3*e - 36*b*c*d^2*e^2 + 21*b^2*d*e^3)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 33.8217, size = 418, normalized size = 2.88 \[ \begin{cases} - \frac{\frac{2 b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{4 b c d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{4 b c \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 c^{2} d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{4}} + \frac{2 c^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}}}{e} & \text{for}\: e \neq 0 \\\frac{\frac{b^{2} x^{3}}{3} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5}}{\sqrt{d}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-(2*b**2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)
/3)/e**2 + 2*b**2*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/
2) - (d + e*x)**(5/2)/5)/e**2 + 4*b*c*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e
*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 + 4*b*c*(d**4/sqrt(d + e*x)
+ 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d +
e*x)**(7/2)/7)/e**3 + 2*c**2*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d
**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 2*c**
2*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d
**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4)/e, Ne(
e, 0)), ((b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5)/sqrt(d), True))

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GIAC/XCAS [A]  time = 0.206899, size = 259, normalized size = 1.79 \[ \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} b^{2} e^{\left (-10\right )} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} b c e^{\left (-21\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} c^{2} e^{\left (-36\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(21*(3*(x*e + d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d
^2*e^8)*b^2*e^(-10) + 18*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 3
5*(x*e + d)^(3/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*e^18)*b*c*e^(-21) + (35*(x*e +
d)^(9/2)*e^32 - 180*(x*e + d)^(7/2)*d*e^32 + 378*(x*e + d)^(5/2)*d^2*e^32 - 420
*(x*e + d)^(3/2)*d^3*e^32 + 315*sqrt(x*e + d)*d^4*e^32)*c^2*e^(-36))*e^(-1)