3.23.79 \(\int \frac {(a+b x+c x^2)^2}{(d+e x)^{3/2}} \, dx\) [2279]

Optimal. Leaf size=162 \[ -\frac {2 \left (c d^2-b d e+a e^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2}}{3 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712} \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^5}-\frac {4 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac {2 \left (a e^2-b d e+c d^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^2/(d + e*x)^(3/2),x]

[Out]

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

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Mathematica [A]
time = 0.12, size = 171, normalized size = 1.06 \begin {gather*} \frac {-6 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )-70 e^2 \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )+28 c e \left (5 a e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 b \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{105 e^5 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^2/(d + e*x)^(3/2),x]

[Out]

(-6*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) - 70*e^2*(3*a^2*e^2 - 6*a*b*e*(2*d +
e*x) + b^2*(8*d^2 + 4*d*e*x - e^2*x^2)) + 28*c*e*(5*a*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + 3*b*(16*d^3 + 8*d^2*e*
x - 2*d*e^2*x^2 + e^3*x^3)))/(105*e^5*Sqrt[d + e*x])

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Maple [A]
time = 0.84, size = 236, normalized size = 1.46

method	result	size
risch	\(\frac {2 \left (15 c^{2} e^{3} x^{3}+42 b c \,e^{3} x^{2}-39 c^{2} d \,e^{2} x^{2}+70 a c \,e^{3} x +35 b^{2} e^{3} x -126 b c d \,e^{2} x +87 c^{2} d^{2} e x +210 a b \,e^{3}-350 a d \,e^{2} c -175 b^{2} d \,e^{2}+462 d^{2} e b c -279 c^{2} d^{3}\right ) \sqrt {e x +d}}{105 e^{5}}-\frac {2 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}\right )}{e^{5} \sqrt {e x +d}}\)	\(187\)
gosper	\(-\frac {2 \left (-15 c^{2} x^{4} e^{4}-42 b c \,e^{4} x^{3}+24 c^{2} d \,e^{3} x^{3}-70 a c \,e^{4} x^{2}-35 b^{2} e^{4} x^{2}+84 b c d \,e^{3} x^{2}-48 c^{2} d^{2} e^{2} x^{2}-210 a b \,e^{4} x +280 a c d \,e^{3} x +140 b^{2} d \,e^{3} x -336 b c \,d^{2} e^{2} x +192 c^{2} d^{3} e x +105 a^{2} e^{4}-420 d \,e^{3} a b +560 a c \,d^{2} e^{2}+280 b^{2} d^{2} e^{2}-672 d^{3} e b c +384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\)	\(194\)
trager	\(-\frac {2 \left (-15 c^{2} x^{4} e^{4}-42 b c \,e^{4} x^{3}+24 c^{2} d \,e^{3} x^{3}-70 a c \,e^{4} x^{2}-35 b^{2} e^{4} x^{2}+84 b c d \,e^{3} x^{2}-48 c^{2} d^{2} e^{2} x^{2}-210 a b \,e^{4} x +280 a c d \,e^{3} x +140 b^{2} d \,e^{3} x -336 b c \,d^{2} e^{2} x +192 c^{2} d^{3} e x +105 a^{2} e^{4}-420 d \,e^{3} a b +560 a c \,d^{2} e^{2}+280 b^{2} d^{2} e^{2}-672 d^{3} e b c +384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\)	\(194\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 b c d e \left (e x +d \right )^{\frac {3}{2}}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a b \,e^{3} \sqrt {e x +d}-8 a c d \,e^{2} \sqrt {e x +d}-4 b^{2} d \,e^{2} \sqrt {e x +d}+12 b c \,d^{2} e \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(236\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 b c d e \left (e x +d \right )^{\frac {3}{2}}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a b \,e^{3} \sqrt {e x +d}-8 a c d \,e^{2} \sqrt {e x +d}-4 b^{2} d \,e^{2} \sqrt {e x +d}+12 b c \,d^{2} e \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^2/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 191, normalized size = 1.18 \begin {gather*} \frac {2}{105} \, {\left ({\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} - 42 \, {\left (2 \, c^{2} d - b c e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 35 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}} - 210 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {105 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d^{2} + a^{2} e^{4}\right )} e^{\left (-4\right )}}{\sqrt {x e + d}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/105*((15*(x*e + d)^(7/2)*c^2 - 42*(2*c^2*d - b*c*e)*(x*e + d)^(5/2) + 35*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2 +
2*a*c*e^2)*(x*e + d)^(3/2) - 210*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2*e^2 + 2*a*c*e^2)*d)*sqrt(x*e + d))*
e^(-4) - 105*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + (b^2*e^2 + 2*a*c*e^2)*d^2 + a^2*e^4)*e^(-4)/sqrt(x*e + d))
*e^(-1)

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Fricas [A]
time = 3.32, size = 170, normalized size = 1.05 \begin {gather*} -\frac {2 \, {\left (384 \, c^{2} d^{4} - {\left (15 \, c^{2} x^{4} + 42 \, b c x^{3} + 210 \, a b x + 35 \, {\left (b^{2} + 2 \, a c\right )} x^{2} - 105 \, a^{2}\right )} e^{4} + 4 \, {\left (6 \, c^{2} d x^{3} + 21 \, b c d x^{2} - 105 \, a b d + 35 \, {\left (b^{2} + 2 \, a c\right )} d x\right )} e^{3} - 8 \, {\left (6 \, c^{2} d^{2} x^{2} + 42 \, b c d^{2} x - 35 \, {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} e^{2} + 96 \, {\left (2 \, c^{2} d^{3} x - 7 \, b c d^{3}\right )} e\right )} \sqrt {x e + d}}{105 \, {\left (x e^{6} + d e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

-2/105*(384*c^2*d^4 - (15*c^2*x^4 + 42*b*c*x^3 + 210*a*b*x + 35*(b^2 + 2*a*c)*x^2 - 105*a^2)*e^4 + 4*(6*c^2*d*
x^3 + 21*b*c*d*x^2 - 105*a*b*d + 35*(b^2 + 2*a*c)*d*x)*e^3 - 8*(6*c^2*d^2*x^2 + 42*b*c*d^2*x - 35*(b^2 + 2*a*c
)*d^2)*e^2 + 96*(2*c^2*d^3*x - 7*b*c*d^3)*e)*sqrt(x*e + d)/(x*e^6 + d*e^5)

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Sympy [A]
time = 17.40, size = 182, normalized size = 1.12 \begin {gather*} \frac {2 c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (4 b c e - 8 c^{2} d\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (4 a c e^{2} + 2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (4 a b e^{3} - 8 a c d e^{2} - 4 b^{2} d e^{2} + 12 b c d^{2} e - 8 c^{2} d^{3}\right )}{e^{5}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{5} \sqrt {d + e x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**(3/2),x)

[Out]

2*c**2*(d + e*x)**(7/2)/(7*e**5) + (d + e*x)**(5/2)*(4*b*c*e - 8*c**2*d)/(5*e**5) + (d + e*x)**(3/2)*(4*a*c*e*
*2 + 2*b**2*e**2 - 12*b*c*d*e + 12*c**2*d**2)/(3*e**5) + sqrt(d + e*x)*(4*a*b*e**3 - 8*a*c*d*e**2 - 4*b**2*d*e
**2 + 12*b*c*d**2*e - 8*c**2*d**3)/e**5 - 2*(a*e**2 - b*d*e + c*d**2)**2/(e**5*sqrt(d + e*x))

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Giac [A]
time = 1.11, size = 253, normalized size = 1.56 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} e^{30} - 84 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d e^{30} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt {x e + d} c^{2} d^{3} e^{30} + 42 \, {\left (x e + d\right )}^{\frac {5}{2}} b c e^{31} - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} b c d e^{31} + 630 \, \sqrt {x e + d} b c d^{2} e^{31} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} e^{32} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a c e^{32} - 210 \, \sqrt {x e + d} b^{2} d e^{32} - 420 \, \sqrt {x e + d} a c d e^{32} + 210 \, \sqrt {x e + d} a b e^{33}\right )} e^{\left (-35\right )} - \frac {2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} e^{\left (-5\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*c^2*e^30 - 84*(x*e + d)^(5/2)*c^2*d*e^30 + 210*(x*e + d)^(3/2)*c^2*d^2*e^30 - 420*sq
rt(x*e + d)*c^2*d^3*e^30 + 42*(x*e + d)^(5/2)*b*c*e^31 - 210*(x*e + d)^(3/2)*b*c*d*e^31 + 630*sqrt(x*e + d)*b*
c*d^2*e^31 + 35*(x*e + d)^(3/2)*b^2*e^32 + 70*(x*e + d)^(3/2)*a*c*e^32 - 210*sqrt(x*e + d)*b^2*d*e^32 - 420*sq
rt(x*e + d)*a*c*d*e^32 + 210*sqrt(x*e + d)*a*b*e^33)*e^(-35) - 2*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*
d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*e^(-5)/sqrt(x*e + d)

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Mupad [B]
time = 0.83, size = 184, normalized size = 1.14 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{3\,e^5}-\frac {2\,a^2\,e^4-4\,a\,b\,d\,e^3+4\,a\,c\,d^2\,e^2+2\,b^2\,d^2\,e^2-4\,b\,c\,d^3\,e+2\,c^2\,d^4}{e^5\,\sqrt {d+e\,x}}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^2/(d + e*x)^(3/2),x)

[Out]

(2*c^2*(d + e*x)^(7/2))/(7*e^5) + ((d + e*x)^(3/2)*(2*b^2*e^2 + 12*c^2*d^2 + 4*a*c*e^2 - 12*b*c*d*e))/(3*e^5)
- (2*a^2*e^4 + 2*c^2*d^4 + 2*b^2*d^2*e^2 - 4*a*b*d*e^3 - 4*b*c*d^3*e + 4*a*c*d^2*e^2)/(e^5*(d + e*x)^(1/2)) -
((8*c^2*d - 4*b*c*e)*(d + e*x)^(5/2))/(5*e^5) + (4*(b*e - 2*c*d)*(d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e))/e^5

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