3.4.16 \(\int (d+e x)^{5/2} (b x+c x^2) \, dx\)

Optimal. Leaf size=68 \[ -\frac {2 (d+e x)^{9/2} (2 c d-b e)}{9 e^3}+\frac {2 d (d+e x)^{7/2} (c d-b e)}{7 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3} \]

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Rubi [A]  time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \begin {gather*} -\frac {2 (d+e x)^{9/2} (2 c d-b e)}{9 e^3}+\frac {2 d (d+e x)^{7/2} (c d-b e)}{7 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(b*x + c*x^2),x]

[Out]

(2*d*(c*d - b*e)*(d + e*x)^(7/2))/(7*e^3) - (2*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^3) + (2*c*(d + e*x)^(11/2))
/(11*e^3)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^{5/2} \left (b x+c x^2\right ) \, dx &=\int \left (\frac {d (c d-b e) (d+e x)^{5/2}}{e^2}+\frac {(-2 c d+b e) (d+e x)^{7/2}}{e^2}+\frac {c (d+e x)^{9/2}}{e^2}\right ) \, dx\\ &=\frac {2 d (c d-b e) (d+e x)^{7/2}}{7 e^3}-\frac {2 (2 c d-b e) (d+e x)^{9/2}}{9 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 50, normalized size = 0.74 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (11 b e (7 e x-2 d)+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(7/2)*(11*b*e*(-2*d + 7*e*x) + c*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)

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IntegrateAlgebraic [A]  time = 0.05, size = 56, normalized size = 0.82 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (77 b e (d+e x)-99 b d e+99 c d^2-154 c d (d+e x)+63 c (d+e x)^2\right )}{693 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(5/2)*(b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(7/2)*(99*c*d^2 - 99*b*d*e - 154*c*d*(d + e*x) + 77*b*e*(d + e*x) + 63*c*(d + e*x)^2))/(693*e^3)

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fricas [B]  time = 0.41, size = 118, normalized size = 1.74 \begin {gather*} \frac {2 \, {\left (63 \, c e^{5} x^{5} + 8 \, c d^{5} - 22 \, b d^{4} e + 7 \, {\left (23 \, c d e^{4} + 11 \, b e^{5}\right )} x^{4} + {\left (113 \, c d^{2} e^{3} + 209 \, b d e^{4}\right )} x^{3} + 3 \, {\left (c d^{3} e^{2} + 55 \, b d^{2} e^{3}\right )} x^{2} - {\left (4 \, c d^{4} e - 11 \, b d^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x),x, algorithm="fricas")

[Out]

2/693*(63*c*e^5*x^5 + 8*c*d^5 - 22*b*d^4*e + 7*(23*c*d*e^4 + 11*b*e^5)*x^4 + (113*c*d^2*e^3 + 209*b*d*e^4)*x^3
+ 3*(c*d^3*e^2 + 55*b*d^2*e^3)*x^2 - (4*c*d^4*e - 11*b*d^3*e^2)*x)*sqrt(e*x + d)/e^3

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giac [B]  time = 0.19, size = 441, normalized size = 6.49 \begin {gather*} \frac {2}{3465} \, {\left (1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b d^{3} e^{\left (-1\right )} + 231 \, {\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^{3} e^{\left (-2\right )} + 693 \, {\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 )} b d^{2} e^{\left (-1\right )} + 297 \, {\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^{2} e^{\left (-2\right )} + 297 \, {\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 )} b d e^{\left (-1\right )} + 33 \, {\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 d e^{\left (-2\right )} + 11 \, {\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 )} b e^{\left (-1\right )} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c e^{\left (-2\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x),x, algorithm="giac")

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*b*d^3*e^(-1) + 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*
d + 15*sqrt(x*e + d)*d^2)*c*d^3*e^(-2) + 693*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)
*b*d^2*e^(-1) + 297*(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^2*e^(-2) + 297*(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)
*b*d*e^(-1) + 33*(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*d*e^(-2) + 11*(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)*b*e^(-1) + 5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)
*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*c*
e^(-2))*e^(-1)

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maple [A]  time = 0.05, size = 47, normalized size = 0.69 \begin {gather*} -\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (-63 c \,e^{2} x^{2}-77 b \,e^{2} x +28 c d e x +22 b d e -8 c \,d^{2}\right )}{693 e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(c*x^2+b*x),x)

[Out]

-2/693*(e*x+d)^(7/2)*(-63*c*e^2*x^2-77*b*e^2*x+28*c*d*e*x+22*b*d*e-8*c*d^2)/e^3

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maxima [A]  time = 1.39, size = 54, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} c - 77 \, {\left (2 \, c d - b e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 99 \, {\left (c d^{2} - b d e\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{693 \, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x),x, algorithm="maxima")

[Out]

2/693*(63*(e*x + d)^(11/2)*c - 77*(2*c*d - b*e)*(e*x + d)^(9/2) + 99*(c*d^2 - b*d*e)*(e*x + d)^(7/2))/e^3

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mupad [B]  time = 0.20, size = 52, normalized size = 0.76 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (63\,c\,{\left (d+e\,x\right )}^2+99\,c\,d^2+77\,b\,e\,\left (d+e\,x\right )-154\,c\,d\,\left (d+e\,x\right )-99\,b\,d\,e\right )}{693\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)*(d + e*x)^(5/2),x)

[Out]

(2*(d + e*x)^(7/2)*(63*c*(d + e*x)^2 + 99*c*d^2 + 77*b*e*(d + e*x) - 154*c*d*(d + e*x) - 99*b*d*e))/(693*e^3)

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sympy [A]  time = 3.67, size = 245, normalized size = 3.60 \begin {gather*} \begin {cases} - \frac {4 b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 b d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 b d e x^{3} \sqrt {d + e x}}{63} + \frac {2 b e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 c d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 c d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 c d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 c d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 c d e x^{4} \sqrt {d + e x}}{99} + \frac {2 c e^{2} x^{5} \sqrt {d + e x}}{11} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (\frac {b x^{2}}{2} + \frac {c x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)*(c*x**2+b*x),x)

[Out]

Piecewise((-4*b*d**4*sqrt(d + e*x)/(63*e**2) + 2*b*d**3*x*sqrt(d + e*x)/(63*e) + 10*b*d**2*x**2*sqrt(d + e*x)/
21 + 38*b*d*e*x**3*sqrt(d + e*x)/63 + 2*b*e**2*x**4*sqrt(d + e*x)/9 + 16*c*d**5*sqrt(d + e*x)/(693*e**3) - 8*c
*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*c*d**3*x**2*sqrt(d + e*x)/(231*e) + 226*c*d**2*x**3*sqrt(d + e*x)/693 + 4
6*c*d*e*x**4*sqrt(d + e*x)/99 + 2*c*e**2*x**5*sqrt(d + e*x)/11, Ne(e, 0)), (d**(5/2)*(b*x**2/2 + c*x**3/3), Tr
ue))

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