3.337 \(\int (d+e x)^{7/2} (b x+c x^2) \, dx\)

Optimal. Leaf size=68 \[ -\frac {2 (d+e x)^{11/2} (2 c d-b e)}{11 e^3}+\frac {2 d (d+e x)^{9/2} (c d-b e)}{9 e^3}+\frac {2 c (d+e x)^{13/2}}{13 e^3} \]

[Out]

2/9*d*(-b*e+c*d)*(e*x+d)^(9/2)/e^3-2/11*(-b*e+2*c*d)*(e*x+d)^(11/2)/e^3+2/13*c*(e*x+d)^(13/2)/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} \[ -\frac {2 (d+e x)^{11/2} (2 c d-b e)}{11 e^3}+\frac {2 d (d+e x)^{9/2} (c d-b e)}{9 e^3}+\frac {2 c (d+e x)^{13/2}}{13 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)*(b*x + c*x^2),x]

[Out]

(2*d*(c*d - b*e)*(d + e*x)^(9/2))/(9*e^3) - (2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^3) + (2*c*(d + e*x)^(13/2
))/(13*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)^{7/2} \left (b x+c x^2\right ) \, dx &=\int \left (\frac {d (c d-b e) (d+e x)^{7/2}}{e^2}+\frac {(-2 c d+b e) (d+e x)^{9/2}}{e^2}+\frac {c (d+e x)^{11/2}}{e^2}\right ) \, dx\\ &=\frac {2 d (c d-b e) (d+e x)^{9/2}}{9 e^3}-\frac {2 (2 c d-b e) (d+e x)^{11/2}}{11 e^3}+\frac {2 c (d+e x)^{13/2}}{13 e^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 50, normalized size = 0.74 \[ \frac {2 (d+e x)^{9/2} \left (13 b e (9 e x-2 d)+c \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)*(b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(9/2)*(13*b*e*(-2*d + 9*e*x) + c*(8*d^2 - 36*d*e*x + 99*e^2*x^2)))/(1287*e^3)

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fricas [B]  time = 0.80, size = 143, normalized size = 2.10 \[ \frac {2 \, {\left (99 \, c e^{6} x^{6} + 8 \, c d^{6} - 26 \, b d^{5} e + 9 \, {\left (40 \, c d e^{5} + 13 \, b e^{6}\right )} x^{5} + 2 \, {\left (229 \, c d^{2} e^{4} + 221 \, b d e^{5}\right )} x^{4} + 2 \, {\left (106 \, c d^{3} e^{3} + 299 \, b d^{2} e^{4}\right )} x^{3} + 3 \, {\left (c d^{4} e^{2} + 104 \, b d^{3} e^{3}\right )} x^{2} - {\left (4 \, c d^{5} e - 13 \, b d^{4} e^{2}\right )} x\right )} \sqrt {e x + d}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="fricas")

[Out]

2/1287*(99*c*e^6*x^6 + 8*c*d^6 - 26*b*d^5*e + 9*(40*c*d*e^5 + 13*b*e^6)*x^5 + 2*(229*c*d^2*e^4 + 221*b*d*e^5)*
x^4 + 2*(106*c*d^3*e^3 + 299*b*d^2*e^4)*x^3 + 3*(c*d^4*e^2 + 104*b*d^3*e^3)*x^2 - (4*c*d^5*e - 13*b*d^4*e^2)*x
)*sqrt(e*x + d)/e^3

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giac [B]  time = 0.20, size = 618, normalized size = 9.09 \[ \frac {2}{45045} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b d^{4} e^{\left (-1\right )} + 3003 \, {\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^{4} e^{\left (-2\right )} + 12012 \, {\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^{3} e^{\left (-1\right )} + 5148 \, {\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^{3} e^{\left (-2\right )} + 7722 \, {\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^{2} e^{\left (-1\right )} + 858 \, {\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^{2} e^{\left (-2\right )} + 572 \, {\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 d e^{\left (-1\right )} + 260 \, {\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 d e^{\left (-2\right )} + 65 \, {\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 )} b e^{\left (-1\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c e^{\left (-2\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*b*d^4*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/
2)*d + 15*sqrt(x*e + d)*d^2)*c*d^4*e^(-2) + 12012*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)
*d^2)*b*d^3*e^(-1) + 5148*(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^3*e^(-2) + 7722*(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^2*e^(-1) + 858*(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^2*e^(-2) + 572*(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*d*e^(-1) + 260*(63*(x*e + d)^(11/2) - 38
5*(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*d*e^(-2) + 65*(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)*b*e^(-1) + 15*(231*(x*e + d)^(13/2) - 1
638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006
*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*c*e^(-2))*e^(-1)

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maple [A]  time = 0.12, size = 47, normalized size = 0.69 \[ -\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (-99 c \,e^{2} x^{2}-117 b \,e^{2} x +36 c d e x +26 b d e -8 c \,d^{2}\right )}{1287 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)*(c*x^2+b*x),x)

[Out]

-2/1287*(e*x+d)^(9/2)*(-99*c*e^2*x^2-117*b*e^2*x+36*c*d*e*x+26*b*d*e-8*c*d^2)/e^3

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maxima [A]  time = 1.35, size = 54, normalized size = 0.79 \[ \frac {2 \, {\left (99 \, {\left (e x + d\right )}^{\frac {13}{2}} c - 117 \, {\left (2 \, c d - b e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 143 \, {\left (c d^{2} - b d e\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="maxima")

[Out]

2/1287*(99*(e*x + d)^(13/2)*c - 117*(2*c*d - b*e)*(e*x + d)^(11/2) + 143*(c*d^2 - b*d*e)*(e*x + d)^(9/2))/e^3

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mupad [B]  time = 0.07, size = 52, normalized size = 0.76 \[ \frac {2\,{\left (d+e\,x\right )}^{9/2}\,\left (99\,c\,{\left (d+e\,x\right )}^2+143\,c\,d^2+117\,b\,e\,\left (d+e\,x\right )-234\,c\,d\,\left (d+e\,x\right )-143\,b\,d\,e\right )}{1287\,e^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)*(d + e*x)^(7/2),x)

[Out]

(2*(d + e*x)^(9/2)*(99*c*(d + e*x)^2 + 143*c*d^2 + 117*b*e*(d + e*x) - 234*c*d*(d + e*x) - 143*b*d*e))/(1287*e
^3)

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sympy [A]  time = 8.53, size = 292, normalized size = 4.29 \[ \begin {cases} - \frac {4 b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 b d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 c d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 c d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 c d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 c d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 c d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 c d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 c e^{3} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {b x^{2}}{2} + \frac {c x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)*(c*x**2+b*x),x)

[Out]

Piecewise((-4*b*d**5*sqrt(d + e*x)/(99*e**2) + 2*b*d**4*x*sqrt(d + e*x)/(99*e) + 16*b*d**3*x**2*sqrt(d + e*x)/
33 + 92*b*d**2*e*x**3*sqrt(d + e*x)/99 + 68*b*d*e**2*x**4*sqrt(d + e*x)/99 + 2*b*e**3*x**5*sqrt(d + e*x)/11 +
16*c*d**6*sqrt(d + e*x)/(1287*e**3) - 8*c*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*c*d**4*x**2*sqrt(d + e*x)/(429*
e) + 424*c*d**3*x**3*sqrt(d + e*x)/1287 + 916*c*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*c*d*e**2*x**5*sqrt(d + e*x
)/143 + 2*c*e**3*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**(7/2)*(b*x**2/2 + c*x**3/3), True))

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