3.339 \(\int (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx\)

Optimal. Leaf size=68 \[ -\frac{2 (d+e x)^{7/2} (2 c d-b e)}{7 e^3}+\frac{2 d (d+e x)^{5/2} (c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0915926, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 (d+e x)^{7/2} (2 c d-b e)}{7 e^3}+\frac{2 d (d+e x)^{5/2} (c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.5053, size = 63, normalized size = 0.93 \[ \frac{2 c \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}} \left (b e - c d\right )}{5 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )}{7 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0562658, size = 50, normalized size = 0.74 \[ \frac{2 (d+e x)^{5/2} \left (9 b e (5 e x-2 d)+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(9*b*e*(-2*d + 5*e*x) + c*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(
315*e^3)

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Maple [A]  time = 0.007, size = 47, normalized size = 0.7 \[ -{\frac{-70\,c{e}^{2}{x}^{2}-90\,b{e}^{2}x+40\,cdex+36\,bde-16\,c{d}^{2}}{315\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(e*x+d)^(5/2)*(-35*c*e^2*x^2-45*b*e^2*x+20*c*d*e*x+18*b*d*e-8*c*d^2)/e^3

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Maxima [A]  time = 0.692031, size = 73, normalized size = 1.07 \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} c - 45 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (c d^{2} - b d e\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(e*x + d)^(9/2)*c - 45*(2*c*d - b*e)*(e*x + d)^(7/2) + 63*(c*d^2 - b*d
*e)*(e*x + d)^(5/2))/e^3

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Fricas [A]  time = 0.215894, size = 128, normalized size = 1.88 \[ \frac{2 \,{\left (35 \, c e^{4} x^{4} + 8 \, c d^{4} - 18 \, b d^{3} e + 5 \,{\left (10 \, c d e^{3} + 9 \, b e^{4}\right )} x^{3} + 3 \,{\left (c d^{2} e^{2} + 24 \, b d e^{3}\right )} x^{2} -{\left (4 \, c d^{3} e - 9 \, b d^{2} e^{2}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*c*e^4*x^4 + 8*c*d^4 - 18*b*d^3*e + 5*(10*c*d*e^3 + 9*b*e^4)*x^3 + 3*(c
*d^2*e^2 + 24*b*d*e^3)*x^2 - (4*c*d^3*e - 9*b*d^2*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 4.63564, size = 178, normalized size = 2.62 \[ \frac{2 b d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 b \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 c d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 c \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.207981, size = 251, normalized size = 3.69 \[ \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b d e^{\left (-1\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} c d e^{\left (-14\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} b e^{\left (-13\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} c e^{\left (-26\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*b*d*e^(-1) + 3*(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)*c*d*e^(-1
4) + 3*(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)*b*e^(-13) + (35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 1
89*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*c*e^(-26))*e^(-1)