3.4.48 \(\int \sqrt {d+e x} (b x+c x^2)^2 \, dx\) [348]

Optimal. Leaf size=147 \[ \frac {2 d^2 (c d-b e)^2 (d+e x)^{3/2}}{3 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{9/2}}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5} \]

[Out]

2/3*d^2*(-b*e+c*d)^2*(e*x+d)^(3/2)/e^5-4/5*d*(-b*e+c*d)*(-b*e+2*c*d)*(e*x+d)^(5/2)/e^5+2/7*(b^2*e^2-6*b*c*d*e+
6*c^2*d^2)*(e*x+d)^(7/2)/e^5-4/9*c*(-b*e+2*c*d)*(e*x+d)^(9/2)/e^5+2/11*c^2*(e*x+d)^(11/2)/e^5

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Rubi [A]
time = 0.04, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {712} \begin {gather*} \frac {2 (d+e x)^{7/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac {2 d^2 (d+e x)^{3/2} (c d-b e)^2}{3 e^5}-\frac {4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}-\frac {4 d (d+e x)^{5/2} (c d-b e) (2 c d-b e)}{5 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*(b*x + c*x^2)^2,x]

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2))/(3*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^5) + (2*(
6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(7/2))/(7*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^5) + (2*c
^2*(d + e*x)^(11/2))/(11*e^5)

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 \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac {d^2 (c d-b e)^2 \sqrt {d+e x}}{e^4}+\frac {2 d (c d-b e) (-2 c d+b e) (d+e x)^{3/2}}{e^4}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{7/2}}{e^4}+\frac {c^2 (d+e x)^{9/2}}{e^4}\right ) \, dx\\ &=\frac {2 d^2 (c d-b e)^2 (d+e x)^{3/2}}{3 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{9/2}}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 124, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (33 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 b c e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(33*b^2*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 22*b*c*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2
+ 35*e^3*x^3) + c^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)))/(3465*e^5)

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Maple [A]
time = 0.43, size = 144, normalized size = 0.98

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 c^{2} x^{4} e^{4}+770 b c \,e^{4} x^{3}-280 c^{2} d \,e^{3} x^{3}+495 b^{2} e^{4} x^{2}-660 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}-396 b^{2} d \,e^{3} x +528 b c \,d^{2} e^{2} x -192 c^{2} d^{3} e x +264 d^{2} e^{2} b^{2}-352 b c \,d^{3} e +128 c^{2} d^{4}\right )}{3465 e^{5}}\)	\(141\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (d^{2} c^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\)	\(144\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (d^{2} c^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\)	\(144\)
trager	\(\frac {2 \left (315 c^{2} e^{5} x^{5}+770 b c \,e^{5} x^{4}+35 c^{2} d \,e^{4} x^{4}+495 b^{2} e^{5} x^{3}+110 b c d \,e^{4} x^{3}-40 c^{2} d^{2} e^{3} x^{3}+99 b^{2} d \,e^{4} x^{2}-132 b c \,d^{2} e^{3} x^{2}+48 c^{2} d^{3} e^{2} x^{2}-132 b^{2} d^{2} e^{3} x +176 b c \,d^{3} e^{2} x -64 c^{2} d^{4} e x +264 b^{2} d^{3} e^{2}-352 b c \,d^{4} e +128 c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\)	\(182\)
risch	\(\frac {2 \left (315 c^{2} e^{5} x^{5}+770 b c \,e^{5} x^{4}+35 c^{2} d \,e^{4} x^{4}+495 b^{2} e^{5} x^{3}+110 b c d \,e^{4} x^{3}-40 c^{2} d^{2} e^{3} x^{3}+99 b^{2} d \,e^{4} x^{2}-132 b c \,d^{2} e^{3} x^{2}+48 c^{2} d^{3} e^{2} x^{2}-132 b^{2} d^{2} e^{3} x +176 b c \,d^{3} e^{2} x -64 c^{2} d^{4} e x +264 b^{2} d^{3} e^{2}-352 b c \,d^{4} e +128 c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/11*c^2*(e*x+d)^(11/2)+1/9*(-2*c^2*d+2*(b*e-c*d)*c)*(e*x+d)^(9/2)+1/7*(d^2*c^2-4*d*(b*e-c*d)*c+(b*e-c*
d)^2)*(e*x+d)^(7/2)+1/5*(2*d^2*(b*e-c*d)*c-2*d*(b*e-c*d)^2)*(e*x+d)^(5/2)+1/3*d^2*(b*e-c*d)^2*(e*x+d)^(3/2))

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Maxima [A]
time = 0.30, size = 144, normalized size = 0.98 \begin {gather*} \frac {2}{3465} \, {\left (315 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{2} - 770 \, {\left (2 \, c^{2} d - b c e\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 495 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}} - 1386 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 1155 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

2/3465*(315*(x*e + d)^(11/2)*c^2 - 770*(2*c^2*d - b*c*e)*(x*e + d)^(9/2) + 495*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^
2)*(x*e + d)^(7/2) - 1386*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(x*e + d)^(5/2) + 1155*(c^2*d^4 - 2*b*c*d^3*e
+ b^2*d^2*e^2)*(x*e + d)^(3/2))*e^(-5)

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Fricas [A]
time = 1.43, size = 167, normalized size = 1.14 \begin {gather*} \frac {2}{3465} \, {\left (128 \, c^{2} d^{5} + 5 \, {\left (63 \, c^{2} x^{5} + 154 \, b c x^{4} + 99 \, b^{2} x^{3}\right )} e^{5} + {\left (35 \, c^{2} d x^{4} + 110 \, b c d x^{3} + 99 \, b^{2} d x^{2}\right )} e^{4} - 4 \, {\left (10 \, c^{2} d^{2} x^{3} + 33 \, b c d^{2} x^{2} + 33 \, b^{2} d^{2} x\right )} e^{3} + 8 \, {\left (6 \, c^{2} d^{3} x^{2} + 22 \, b c d^{3} x + 33 \, b^{2} d^{3}\right )} e^{2} - 32 \, {\left (2 \, c^{2} d^{4} x + 11 \, b c d^{4}\right )} e\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

2/3465*(128*c^2*d^5 + 5*(63*c^2*x^5 + 154*b*c*x^4 + 99*b^2*x^3)*e^5 + (35*c^2*d*x^4 + 110*b*c*d*x^3 + 99*b^2*d
*x^2)*e^4 - 4*(10*c^2*d^2*x^3 + 33*b*c*d^2*x^2 + 33*b^2*d^2*x)*e^3 + 8*(6*c^2*d^3*x^2 + 22*b*c*d^3*x + 33*b^2*
d^3)*e^2 - 32*(2*c^2*d^4*x + 11*b*c*d^4)*e)*sqrt(x*e + d)*e^(-5)

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Sympy [A]
time = 1.70, size = 173, normalized size = 1.18 \begin {gather*} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(c*x**2+b*x)**2,x)

[Out]

2*(c**2*(d + e*x)**(11/2)/(11*e**4) + (d + e*x)**(9/2)*(2*b*c*e - 4*c**2*d)/(9*e**4) + (d + e*x)**(7/2)*(b**2*
e**2 - 6*b*c*d*e + 6*c**2*d**2)/(7*e**4) + (d + e*x)**(5/2)*(-2*b**2*d*e**2 + 6*b*c*d**2*e - 4*c**2*d**3)/(5*e
**4) + (d + e*x)**(3/2)*(b**2*d**2*e**2 - 2*b*c*d**3*e + c**2*d**4)/(3*e**4))/e

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (131) = 262\).
time = 1.16, size = 375, normalized size = 2.55 \begin {gather*} \frac {2}{3465} \, {\left (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 )} b^{2} d e^{\left (-2\right )} + 198 \, {\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 c d e^{\left (-3\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 )} c^{2} d e^{\left (-4\right )} + 99 \, {\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^{2} e^{\left (-2\right )} + 22 \, {\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 c e^{\left (-3\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^{2} e^{\left (-4\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^2*d*e^(-2) + 198*(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*c*d*e^(-3) + 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)*c^2
*d*e^(-4) + 99*(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^2*
e^(-2) + 22*(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*c*e^(-3) + 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^2*e^(-4))*e^(-1)

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Mupad [B]
time = 0.04, size = 138, normalized size = 0.94 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{5\,e^5}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{7\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2*(d + e*x)^(1/2),x)

[Out]

(2*c^2*(d + e*x)^(11/2))/(11*e^5) - ((d + e*x)^(5/2)*(8*c^2*d^3 + 4*b^2*d*e^2 - 12*b*c*d^2*e))/(5*e^5) + ((d +
e*x)^(7/2)*(2*b^2*e^2 + 12*c^2*d^2 - 12*b*c*d*e))/(7*e^5) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(9/2))/(9*e^5) + (
2*d^2*(b*e - c*d)^2*(d + e*x)^(3/2))/(3*e^5)

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