3.15.6 \(\int \frac {(b+2 c x) (a+b x+c x^2)}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=130 \[ \frac {2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^4}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac {6 c (d+e x)^{5/2} (2 c d-b e)}{5 e^4}+\frac {4 c^2 (d+e x)^{7/2}}{7 e^4} \]

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Rubi [A]  time = 0.06, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \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^4}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac {6 c (d+e x)^{5/2} (2 c d-b e)}{5 e^4}+\frac {4 c^2 (d+e x)^{7/2}}{7 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x])/e^4 + (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^4) - (6*c*(2*c*d - b*e)*(d + e*x)^(5/2))/(5*e^4) + (4*c^2*(d + e*x)^(7/2))/(7*e^4)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 109, normalized size = 0.84 \begin {gather*} \frac {2 \sqrt {d+e x} \left (7 c e \left (10 a e (e x-2 d)+3 b \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+35 b e^2 (3 a e-2 b d+b e x)-6 c^2 \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )}{105 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(35*b*e^2*(-2*b*d + 3*a*e + b*e*x) - 6*c^2*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3) + 7
*c*e*(10*a*e*(-2*d + e*x) + 3*b*(8*d^2 - 4*d*e*x + 3*e^2*x^2))))/(105*e^4)

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IntegrateAlgebraic [A]  time = 0.08, size = 143, normalized size = 1.10 \begin {gather*} \frac {2 \sqrt {d+e x} \left (105 a b e^3+70 a c e^2 (d+e x)-210 a c d e^2+35 b^2 e^2 (d+e x)-105 b^2 d e^2+315 b c d^2 e-210 b c d e (d+e x)+63 b c e (d+e x)^2-210 c^2 d^3+210 c^2 d^2 (d+e x)-126 c^2 d (d+e x)^2+30 c^2 (d+e x)^3\right )}{105 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-210*c^2*d^3 + 315*b*c*d^2*e - 105*b^2*d*e^2 - 210*a*c*d*e^2 + 105*a*b*e^3 + 210*c^2*d^2*(d
+ e*x) - 210*b*c*d*e*(d + e*x) + 35*b^2*e^2*(d + e*x) + 70*a*c*e^2*(d + e*x) - 126*c^2*d*(d + e*x)^2 + 63*b*c*
e*(d + e*x)^2 + 30*c^2*(d + e*x)^3))/(105*e^4)

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fricas [A]  time = 0.39, size = 116, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (30 \, c^{2} e^{3} x^{3} - 96 \, c^{2} d^{3} + 168 \, b c d^{2} e + 105 \, a b e^{3} - 70 \, {\left (b^{2} + 2 \, a c\right )} d e^{2} - 9 \, {\left (4 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} x^{2} + {\left (48 \, c^{2} d^{2} e - 84 \, b c d e^{2} + 35 \, {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(30*c^2*e^3*x^3 - 96*c^2*d^3 + 168*b*c*d^2*e + 105*a*b*e^3 - 70*(b^2 + 2*a*c)*d*e^2 - 9*(4*c^2*d*e^2 - 7
*b*c*e^3)*x^2 + (48*c^2*d^2*e - 84*b*c*d*e^2 + 35*(b^2 + 2*a*c)*e^3)*x)*sqrt(e*x + d)/e^4

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giac [A]  time = 0.27, size = 166, normalized size = 1.28 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b^{2} e^{\left (-1\right )} + 70 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a c e^{\left (-1\right )} + 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 c e^{\left (-2\right )} + 6 \, {\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^{2} e^{\left (-3\right )} + 105 \, \sqrt {x e + d} a b\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*b^2*e^(-1) + 70*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*c*e^(-
1) + 21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b*c*e^(-2) + 6*(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^2*e^(-3) + 105*sqrt(x*e + d)*a*b)*e^(-
1)

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maple [A]  time = 0.05, size = 123, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (30 c^{2} x^{3} e^{3}+63 b c \,e^{3} x^{2}-36 c^{2} d \,e^{2} x^{2}+70 a c \,e^{3} x +35 b^{2} e^{3} x -84 b c d \,e^{2} x +48 c^{2} d^{2} e x +105 a b \,e^{3}-140 a c d \,e^{2}-70 b^{2} d \,e^{2}+168 b c \,d^{2} e -96 c^{2} d^{3}\right )}{105 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(e*x+d)^(1/2)*(30*c^2*e^3*x^3+63*b*c*e^3*x^2-36*c^2*d*e^2*x^2+70*a*c*e^3*x+35*b^2*e^3*x-84*b*c*d*e^2*x+4
8*c^2*d^2*e*x+105*a*b*e^3-140*a*c*d*e^2-70*b^2*d*e^2+168*b*c*d^2*e-96*c^2*d^3)/e^4

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maxima [A]  time = 0.53, size = 121, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (30 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} - 63 \, {\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}} - 105 \, {\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}\right )}}{105 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(30*(e*x + d)^(7/2)*c^2 - 63*(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) - 105*(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

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mupad [B]  time = 0.07, size = 118, normalized size = 0.91 \begin {gather*} \frac {4\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\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^4}-\frac {\left (12\,c^2\,d-6\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 45.95, size = 427, normalized size = 3.28 \begin {gather*} \begin {cases} \frac {- \frac {2 a b d}{\sqrt {d + e x}} - 2 a b \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {4 a c d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {4 a c \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} - \frac {2 b^{2} d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 b^{2} \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} - \frac {6 b c 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 {6 b c \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 c^{2} 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 c^{2} \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}}}{e} & \text {for}\: e \neq 0 \\\frac {\left (a + b x + c x^{2}\right )^{2}}{2 \sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*a*b*d/sqrt(d + e*x) - 2*a*b*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 4*a*c*d*(-d/sqrt(d + e*x) - sq
rt(d + e*x))/e - 4*a*c*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 2*b**2*d*(-d/sqrt(d +
e*x) - sqrt(d + e*x))/e - 2*b**2*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 6*b*c*d*(d
**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 6*b*c*(-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*c**2*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*c**2*(d**4/sqrt(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**3)/e, Ne(e, 0)), ((a + b*x + c*x**2)**2/(2*s
qrt(d)), True))

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