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

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]

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

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Rubi [A]  time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \[ \frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}{5 e^3}+\frac {2 c (d+e x)^{9/2}}{9 e^3}-\frac {4 c d (d+e x)^{7/2}}{7 e^3} \]

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)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int (d+e x)^{3/2} \left (a+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^{3/2}}{e^2}-\frac {2 c d (d+e x)^{5/2}}{e^2}+\frac {c (d+e x)^{7/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^3}-\frac {4 c d (d+e x)^{7/2}}{7 e^3}+\frac {2 c (d+e x)^{9/2}}{9 e^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 44, normalized size = 0.70 \[ \frac {2 (d+e x)^{5/2} \left (63 a e^2+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]

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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fricas [A]  time = 1.20, size = 85, normalized size = 1.35 \[ \frac {2 \, {\left (35 \, c e^{4} x^{4} + 50 \, c d e^{3} x^{3} + 8 \, c d^{4} + 63 \, a d^{2} e^{2} + 3 \, {\left (c d^{2} e^{2} + 21 \, a e^{4}\right )} x^{2} - 2 \, {\left (2 \, c d^{3} e - 63 \, a d e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+a),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 + 21*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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giac [B]  time = 0.17, size = 243, normalized size = 3.86 \[ \frac {2}{315} \, {\left (21 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c d^{2} e^{\left (-2\right )} + 18 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c d e^{\left (-2\right )} + 315 \, \sqrt {x e + d} a d^{2} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c e^{\left (-2\right )} + 21 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*c*d^2*e^(-2) + 18*(5*(x*e + d)^(7/
2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*c*d*e^(-2) + 315*sqrt(x*e + d)*a*d^
2 + 210*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*d + (35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d
)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*c*e^(-2) + 21*(3*(x*e + d)^(5/2) - 10*(x*e + d)
^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a)*e^(-1)

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maple [A]  time = 0.05, size = 41, normalized size = 0.65 \[ \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (35 c \,e^{2} x^{2}-20 c d e x +63 a \,e^{2}+8 c \,d^{2}\right )}{315 e^{3}} \]

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 = 1.34, size = 47, normalized size = 0.75 \[ \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} c - 90 \, {\left (e x + d\right )}^{\frac {7}{2}} c d + 63 \, {\left (c d^{2} + a e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{315 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+a),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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mupad [B]  time = 0.34, size = 44, normalized size = 0.70 \[ \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (35\,c\,{\left (d+e\,x\right )}^2+63\,a\,e^2+63\,c\,d^2-90\,c\,d\,\left (d+e\,x\right )\right )}{315\,e^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(d + e*x)^(3/2),x)

[Out]

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

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sympy [A]  time = 6.83, 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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