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

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

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Rubi [A]  time = 0.07, 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 = {698} \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 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 \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 = 0.13, size = 171, normalized size = 1.06 \begin {gather*} \frac {-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 )-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 )}{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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IntegrateAlgebraic [A]  time = 0.11, size = 229, normalized size = 1.41 \begin {gather*} \frac {2 \left (-105 a^2 e^4+210 a b e^3 (d+e x)+210 a b d e^3-210 a c d^2 e^2-420 a c d e^2 (d+e x)+70 a c e^2 (d+e x)^2-105 b^2 d^2 e^2-210 b^2 d e^2 (d+e x)+35 b^2 e^2 (d+e x)^2+210 b c d^3 e+630 b c d^2 e (d+e x)-210 b c d e (d+e x)^2+42 b c e (d+e x)^3-105 c^2 d^4-420 c^2 d^3 (d+e x)+210 c^2 d^2 (d+e x)^2-84 c^2 d (d+e x)^3+15 c^2 (d+e x)^4\right )}{105 e^5 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-105*c^2*d^4 + 210*b*c*d^3*e - 105*b^2*d^2*e^2 - 210*a*c*d^2*e^2 + 210*a*b*d*e^3 - 105*a^2*e^4 - 420*c^2*d
^3*(d + e*x) + 630*b*c*d^2*e*(d + e*x) - 210*b^2*d*e^2*(d + e*x) - 420*a*c*d*e^2*(d + e*x) + 210*a*b*e^3*(d +
e*x) + 210*c^2*d^2*(d + e*x)^2 - 210*b*c*d*e*(d + e*x)^2 + 35*b^2*e^2*(d + e*x)^2 + 70*a*c*e^2*(d + e*x)^2 - 8
4*c^2*d*(d + e*x)^3 + 42*b*c*e*(d + e*x)^3 + 15*c^2*(d + e*x)^4))/(105*e^5*Sqrt[d + e*x])

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fricas [A]  time = 0.42, size = 185, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (15 \, c^{2} e^{4} x^{4} - 384 \, c^{2} d^{4} + 672 \, b c d^{3} e + 420 \, a b d e^{3} - 105 \, a^{2} e^{4} - 280 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 6 \, {\left (4 \, c^{2} d e^{3} - 7 \, b c e^{4}\right )} x^{3} + {\left (48 \, c^{2} d^{2} e^{2} - 84 \, b c d e^{3} + 35 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 2 \, {\left (96 \, c^{2} d^{3} e - 168 \, b c d^{2} e^{2} - 105 \, a b e^{4} + 70 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{6} x + 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*(15*c^2*e^4*x^4 - 384*c^2*d^4 + 672*b*c*d^3*e + 420*a*b*d*e^3 - 105*a^2*e^4 - 280*(b^2 + 2*a*c)*d^2*e^2
- 6*(4*c^2*d*e^3 - 7*b*c*e^4)*x^3 + (48*c^2*d^2*e^2 - 84*b*c*d*e^3 + 35*(b^2 + 2*a*c)*e^4)*x^2 - 2*(96*c^2*d^3
*e - 168*b*c*d^2*e^2 - 105*a*b*e^4 + 70*(b^2 + 2*a*c)*d*e^3)*x)*sqrt(e*x + d)/(e^6*x + d*e^5)

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giac [A]  time = 0.23, 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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maple [A]  time = 0.05, size = 194, normalized size = 1.20 \begin {gather*} -\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 a b d \,e^{3}+560 a c \,d^{2} e^{2}+280 b^{2} d^{2} e^{2}-672 b c \,d^{3} e +384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}} \end {gather*}

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)

[Out]

-2/105/(e*x+d)^(1/2)*(-15*c^2*e^4*x^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*a*b*d*e^3+560*a*c*d^2*e^2+280*b^2*d^2*e^2-672*b*c*d^3*e+384*c^2*d^4)/e^5

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maxima [A]  time = 0.83, size = 184, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} - 42 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + 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} + 2 \, a c\right )} d e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {105 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )}}{\sqrt {e x + d} e^{4}}\right )}}{105 \, e} \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*(e*x + d)^(7/2)*c^2 - 42*(2*c^2*d - b*c*e)*(e*x + d)^(5/2) + 35*(6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a
*c)*e^2)*(e*x + d)^(3/2) - 210*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*sqrt(e*x + d))/e^4 -
105*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)/(sqrt(e*x + d)*e^4))/e

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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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sympy [A]  time = 41.91, 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}} \left (4 b c e - 8 c^{2} d\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \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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