3.4.38 \(\int (d+e x)^{5/2} (b x+c x^2) \, dx\) [338]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {712} \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 712

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 (-2 d+7 e x)+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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Maple [A]
time = 0.40, size = 52, normalized size = 0.76

method	result	size
gosper	\(-\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (-63 c \,x^{2} e^{2}-77 b \,e^{2} x +28 c d e x +22 b d e -8 c \,d^{2}\right )}{693 e^{3}}\)	\(47\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {2 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(52\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {2 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(52\)
trager	\(-\frac {2 \left (-63 e^{5} c \,x^{5}-77 b \,e^{5} x^{4}-161 c d \,e^{4} x^{4}-209 b d \,e^{4} x^{3}-113 c \,d^{2} e^{3} x^{3}-165 b \,d^{2} e^{3} x^{2}-3 c \,d^{3} e^{2} x^{2}-11 b \,d^{3} e^{2} x +4 c \,d^{4} e x +22 b \,d^{4} e -8 c \,d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(119\)
risch	\(-\frac {2 \left (-63 e^{5} c \,x^{5}-77 b \,e^{5} x^{4}-161 c d \,e^{4} x^{4}-209 b d \,e^{4} x^{3}-113 c \,d^{2} e^{3} x^{3}-165 b \,d^{2} e^{3} x^{2}-3 c \,d^{3} e^{2} x^{2}-11 b \,d^{3} e^{2} x +4 c \,d^{4} e x +22 b \,d^{4} e -8 c \,d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^3*(1/11*c*(e*x+d)^(11/2)+1/9*(b*e-2*c*d)*(e*x+d)^(9/2)-1/7*d*(b*e-c*d)*(e*x+d)^(7/2))

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Maxima [A]
time = 0.28, size = 58, normalized size = 0.85 \begin {gather*} \frac {2}{693} \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} c - 77 \, {\left (2 \, c d - b e\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 99 \, {\left (c d^{2} - b d e\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-3\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="maxima")

[Out]

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

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Fricas [A]
time = 1.71, size = 114, normalized size = 1.68 \begin {gather*} \frac {2}{693} \, {\left (8 \, c d^{5} + 7 \, {\left (9 \, c x^{5} + 11 \, b x^{4}\right )} e^{5} + {\left (161 \, c d x^{4} + 209 \, b d x^{3}\right )} e^{4} + {\left (113 \, c d^{2} x^{3} + 165 \, b d^{2} x^{2}\right )} e^{3} + {\left (3 \, c d^{3} x^{2} + 11 \, b d^{3} x\right )} e^{2} - 2 \, {\left (2 \, c d^{4} x + 11 \, b d^{4}\right )} e\right )} \sqrt {x e + d} e^{\left (-3\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="fricas")

[Out]

2/693*(8*c*d^5 + 7*(9*c*x^5 + 11*b*x^4)*e^5 + (161*c*d*x^4 + 209*b*d*x^3)*e^4 + (113*c*d^2*x^3 + 165*b*d^2*x^2
)*e^3 + (3*c*d^3*x^2 + 11*b*d^3*x)*e^2 - 2*(2*c*d^4*x + 11*b*d^4)*e)*sqrt(x*e + d)*e^(-3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (65) = 130\).
time = 0.32, 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (58) = 116\).
time = 1.46, 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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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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