3.4.49 \(\int \frac {(b x+c x^2)^2}{\sqrt {d+e x}} \, dx\) [349]

Optimal. Leaf size=145 \[ \frac {2 d^2 (c d-b e)^2 \sqrt {d+e x}}{e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{3/2}}{3 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5} \]

[Out]

-4/3*d*(-b*e+c*d)*(-b*e+2*c*d)*(e*x+d)^(3/2)/e^5+2/5*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)*(e*x+d)^(5/2)/e^5-4/7*c*(-b
*e+2*c*d)*(e*x+d)^(7/2)/e^5+2/9*c^2*(e*x+d)^(9/2)/e^5+2*d^2*(-b*e+c*d)^2*(e*x+d)^(1/2)/e^5

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Rubi [A]
time = 0.04, antiderivative size = 145, 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)^{5/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{5 e^5}+\frac {2 d^2 \sqrt {d+e x} (c d-b e)^2}{e^5}-\frac {4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}-\frac {4 d (d+e x)^{3/2} (c d-b e) (2 c d-b e)}{3 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^2/Sqrt[d + e*x],x]

[Out]

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

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Mathematica [A]
time = 0.06, size = 124, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt {d+e x} \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(21*b^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 18*b*c*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e
^3*x^3) + c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)))/(315*e^5)

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Maple [A]
time = 0.43, size = 143, normalized size = 0.99

method	result	size
gosper	\(\frac {2 \left (35 c^{2} x^{4} e^{4}+90 b c \,e^{4} x^{3}-40 c^{2} d \,e^{3} x^{3}+63 b^{2} e^{4} x^{2}-108 b c d \,e^{3} x^{2}+48 c^{2} d^{2} e^{2} x^{2}-84 b^{2} d \,e^{3} x +144 b c \,d^{2} e^{2} x -64 c^{2} d^{3} e x +168 d^{2} e^{2} b^{2}-288 b c \,d^{3} e +128 c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}}\)	\(141\)
trager	\(\frac {2 \left (35 c^{2} x^{4} e^{4}+90 b c \,e^{4} x^{3}-40 c^{2} d \,e^{3} x^{3}+63 b^{2} e^{4} x^{2}-108 b c d \,e^{3} x^{2}+48 c^{2} d^{2} e^{2} x^{2}-84 b^{2} d \,e^{3} x +144 b c \,d^{2} e^{2} x -64 c^{2} d^{3} e x +168 d^{2} e^{2} b^{2}-288 b c \,d^{3} e +128 c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}}\)	\(141\)
risch	\(\frac {2 \left (35 c^{2} x^{4} e^{4}+90 b c \,e^{4} x^{3}-40 c^{2} d \,e^{3} x^{3}+63 b^{2} e^{4} x^{2}-108 b c d \,e^{3} x^{2}+48 c^{2} d^{2} e^{2} x^{2}-84 b^{2} d \,e^{3} x +144 b c \,d^{2} e^{2} x -64 c^{2} d^{3} e x +168 d^{2} e^{2} b^{2}-288 b c \,d^{3} e +128 c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}}\)	\(141\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (d^{2} c^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}{e^{5}}\)	\(143\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (d^{2} c^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}{e^{5}}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 168, normalized size = 1.16 \begin {gather*} \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 )} b^{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 )} b c e^{\left (-3\right )} + {\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} e^{\left (-4\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^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)*b*c*e^(-3) + (35*(x*e + d)^(9/2) - 18
0*(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*e^(-4))*e
^(-1)

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Fricas [A]
time = 1.75, size = 130, normalized size = 0.90 \begin {gather*} \frac {2}{315} \, {\left (128 \, c^{2} d^{4} + {\left (35 \, c^{2} x^{4} + 90 \, b c x^{3} + 63 \, b^{2} x^{2}\right )} e^{4} - 4 \, {\left (10 \, c^{2} d x^{3} + 27 \, b c d x^{2} + 21 \, b^{2} d x\right )} e^{3} + 24 \, {\left (2 \, c^{2} d^{2} x^{2} + 6 \, b c d^{2} x + 7 \, b^{2} d^{2}\right )} e^{2} - 32 \, {\left (2 \, c^{2} d^{3} x + 9 \, b c d^{3}\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((c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/315*(128*c^2*d^4 + (35*c^2*x^4 + 90*b*c*x^3 + 63*b^2*x^2)*e^4 - 4*(10*c^2*d*x^3 + 27*b*c*d*x^2 + 21*b^2*d*x)
*e^3 + 24*(2*c^2*d^2*x^2 + 6*b*c*d^2*x + 7*b^2*d^2)*e^2 - 32*(2*c^2*d^3*x + 9*b*c*d^3)*e)*sqrt(x*e + d)*e^(-5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (143) = 286\).
time = 22.63, size = 418, normalized size = 2.88 \begin {gather*} \begin {cases} \frac {- \frac {2 b^{2} d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {2 b^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {4 b c d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {4 b c \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 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^{2} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {2 c^{2} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}}}{e} & \text {for}\: e \neq 0 \\\frac {\frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*b**2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 2*b**2*(-d**3/sqrt(
d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 4*b*c*d*(-d**3/sqrt(d + e*x)
- 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 4*b*c*(d**4/sqrt(d + e*x) + 4*d**3*s
qrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*c**2*d*(d**4/sq
rt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e*
*4 - 2*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2)
+ 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4)/e, Ne(e, 0)), ((b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5)/
sqrt(d), True))

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Giac [A]
time = 1.53, size = 168, normalized size = 1.16 \begin {gather*} \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 )} b^{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 )} b c e^{\left (-3\right )} + {\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} e^{\left (-4\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^(1/2),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)*b^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)*b*c*e^(-3) + (35*(x*e + d)^(9/2) - 18
0*(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*e^(-4))*e
^(-1)

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Mupad [B]
time = 0.04, size = 138, normalized size = 0.95 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{3\,e^5}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{5\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}}{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)^(9/2))/(9*e^5) - ((d + e*x)^(3/2)*(8*c^2*d^3 + 4*b^2*d*e^2 - 12*b*c*d^2*e))/(3*e^5) + ((d + e
*x)^(5/2)*(2*b^2*e^2 + 12*c^2*d^2 - 12*b*c*d*e))/(5*e^5) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(7/2))/(7*e^5) + (2*
d^2*(b*e - c*d)^2*(d + e*x)^(1/2))/e^5

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