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

Optimal. Leaf size=63 \[ \frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}{5 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3}-\frac{4 c d (d+e x)^{7/2}}{7 e^3} \]

[Out]

(2*(c*d^2 + a*e^2)*(d + e*x)^(5/2))/(5*e^3) - (4*c*d*(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 = 0.0709492, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}{5 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3}-\frac{4 c d (d+e x)^{7/2}}{7 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*d^2 + a*e^2)*(d + e*x)^(5/2))/(5*e^3) - (4*c*d*(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 = 11.3424, size = 60, normalized size = 0.95 \[ - \frac{4 c d \left (d + e x\right )^{\frac{7}{2}}}{7 e^{3}} + \frac{2 c \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )}{5 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0570446, size = 44, normalized size = 0.7 \[ \frac{2 (d+e x)^{5/2} \left (63 a e^2+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(63*a*e^2 + c*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*e^3)

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Maple [A]  time = 0.005, size = 41, normalized size = 0.7 \[{\frac{70\,c{e}^{2}{x}^{2}-40\,cdex+126\,a{e}^{2}+16\,c{d}^{2}}{315\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(c*x^2+a),x)

[Out]

2/315*(e*x+d)^(5/2)*(35*c*e^2*x^2-20*c*d*e*x+63*a*e^2+8*c*d^2)/e^3

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Maxima [A]  time = 0.693386, size = 63, normalized size = 1. \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} c - 90 \,{\left (e x + d\right )}^{\frac{7}{2}} c d + 63 \,{\left (c d^{2} + a e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(e*x + d)^(9/2)*c - 90*(e*x + d)^(7/2)*c*d + 63*(c*d^2 + a*e^2)*(e*x +
d)^(5/2))/e^3

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Fricas [A]  time = 0.208655, size = 115, normalized size = 1.83 \[ \frac{2 \,{\left (35 \, c e^{4} x^{4} + 50 \, c d e^{3} x^{3} + 8 \, c d^{4} + 63 \, a d^{2} e^{2} + 3 \,{\left (c d^{2} e^{2} + 21 \, a e^{4}\right )} x^{2} - 2 \,{\left (2 \, c d^{3} e - 63 \, a d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*c*e^4*x^4 + 50*c*d*e^3*x^3 + 8*c*d^4 + 63*a*d^2*e^2 + 3*(c*d^2*e^2 + 2
1*a*e^4)*x^2 - 2*(2*c*d^3*e - 63*a*d*e^3)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 4.63215, size = 155, normalized size = 2.46 \[ a d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{2 c d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 c \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a*(-d
*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 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 = 0.212246, size = 201, normalized size = 3.19 \[ \frac{2}{315} \,{\left (3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} c d e^{\left (-14\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a d +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} c e^{\left (-26\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(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^(-14) + 105*(x*e + d)^(3/2)*a*d + (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)*c*e^(-26) + 21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a)*e^(-1)