\(\int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx\) [841]

Optimal result

Integrand size = 29, antiderivative size = 281 \[ \int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac {2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt {f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}+\frac {2 \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^3 (d+e x)^{3/2}}-\frac {4 g \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^4 \sqrt {d+e x}} \]

[Out]

-2/7*(a+d*(-b*e+c*d)/e^2)*(g*x+f)^(1/2)/(-d*g+e*f)/(e*x+d)^(7/2)+2/35*(2*c*d*(-4*d*g+7*e*f)-e*(-6*a*e*g-b*d*g+
7*b*e*f))*(g*x+f)^(1/2)/e^2/(-d*g+e*f)^2/(e*x+d)^(5/2)+2/105*(4*e*g*(-6*a*e*g-b*d*g+7*b*e*f)-c*(3*d^2*g^2-14*d
*e*f*g+35*e^2*f^2))*(g*x+f)^(1/2)/e^2/(-d*g+e*f)^3/(e*x+d)^(3/2)-4/105*g*(4*e*g*(-6*a*e*g-b*d*g+7*b*e*f)-c*(3*
d^2*g^2-14*d*e*f*g+35*e^2*f^2))*(g*x+f)^(1/2)/e^2/(-d*g+e*f)^4/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {963, 79, 47, 37} \[ \int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx=-\frac {4 g \sqrt {f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 \sqrt {d+e x} (e f-d g)^4}+\frac {2 \sqrt {f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 (d+e x)^{3/2} (e f-d g)^3}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{7 (d+e x)^{7/2} (e f-d g)}+\frac {2 \sqrt {f+g x} (2 c d (7 e f-4 d g)-e (-6 a e g-b d g+7 b e f))}{35 e^2 (d+e x)^{5/2} (e f-d g)^2} \]

[In]

Int[(a + b*x + c*x^2)/((d + e*x)^(9/2)*Sqrt[f + g*x]),x]

[Out]

(-2*(a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/(7*(e*f - d*g)*(d + e*x)^(7/2)) + (2*(2*c*d*(7*e*f - 4*d*g) - e*(
7*b*e*f - b*d*g - 6*a*e*g))*Sqrt[f + g*x])/(35*e^2*(e*f - d*g)^2*(d + e*x)^(5/2)) + (2*(4*e*g*(7*b*e*f - b*d*g
- 6*a*e*g) - c*(35*e^2*f^2 - 14*d*e*f*g + 3*d^2*g^2))*Sqrt[f + g*x])/(105*e^2*(e*f - d*g)^3*(d + e*x)^(3/2))
- (4*g*(4*e*g*(7*b*e*f - b*d*g - 6*a*e*g) - c*(35*e^2*f^2 - 14*d*e*f*g + 3*d^2*g^2))*Sqrt[f + g*x])/(105*e^2*(
e*f - d*g)^4*Sqrt[d + e*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 963

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}-\frac {2 \int \frac {\frac {c d (7 e f-d g)-e (7 b e f-b d g-6 a e g)}{2 e^2}-\frac {7}{2} c \left (f-\frac {d g}{e}\right ) x}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx}{7 (e f-d g)} \\ & = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac {2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt {f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}-\frac {\left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx}{35 e^2 (e f-d g)^2} \\ & = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac {2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt {f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}+\frac {2 \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^3 (d+e x)^{3/2}}+\frac {\left (2 g \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right )\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx}{105 e^2 (e f-d g)^3} \\ & = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac {2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt {f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}+\frac {2 \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^3 (d+e x)^{3/2}}-\frac {4 g \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^4 \sqrt {d+e x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.07 \[ \int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx=-\frac {2 \sqrt {f+g x} \left (-105 c f^2 g (d+e x)^3+105 b f g^2 (d+e x)^3-105 a g^3 (d+e x)^3+35 c e f^2 (d+e x)^2 (f+g x)+70 c d f g (d+e x)^2 (f+g x)-70 b e f g (d+e x)^2 (f+g x)-35 b d g^2 (d+e x)^2 (f+g x)+105 a e g^2 (d+e x)^2 (f+g x)-42 c d e f (d+e x) (f+g x)^2+21 b e^2 f (d+e x) (f+g x)^2-21 c d^2 g (d+e x) (f+g x)^2+42 b d e g (d+e x) (f+g x)^2-63 a e^2 g (d+e x) (f+g x)^2+15 c d^2 e (f+g x)^3-15 b d e^2 (f+g x)^3+15 a e^3 (f+g x)^3\right )}{105 (e f-d g)^4 (d+e x)^{7/2}} \]

[In]

Integrate[(a + b*x + c*x^2)/((d + e*x)^(9/2)*Sqrt[f + g*x]),x]

[Out]

(-2*Sqrt[f + g*x]*(-105*c*f^2*g*(d + e*x)^3 + 105*b*f*g^2*(d + e*x)^3 - 105*a*g^3*(d + e*x)^3 + 35*c*e*f^2*(d
+ e*x)^2*(f + g*x) + 70*c*d*f*g*(d + e*x)^2*(f + g*x) - 70*b*e*f*g*(d + e*x)^2*(f + g*x) - 35*b*d*g^2*(d + e*x
)^2*(f + g*x) + 105*a*e*g^2*(d + e*x)^2*(f + g*x) - 42*c*d*e*f*(d + e*x)*(f + g*x)^2 + 21*b*e^2*f*(d + e*x)*(f
+ g*x)^2 - 21*c*d^2*g*(d + e*x)*(f + g*x)^2 + 42*b*d*e*g*(d + e*x)*(f + g*x)^2 - 63*a*e^2*g*(d + e*x)*(f + g*
x)^2 + 15*c*d^2*e*(f + g*x)^3 - 15*b*d*e^2*(f + g*x)^3 + 15*a*e^3*(f + g*x)^3))/(105*(e*f - d*g)^4*(d + e*x)^(
7/2))

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.52

method	result	size
default	\(\frac {2 \sqrt {g x +f}\, \left (48 a \,e^{3} g^{3} x^{3}+8 b d \,e^{2} g^{3} x^{3}-56 b \,e^{3} f \,g^{2} x^{3}+6 c \,d^{2} e \,g^{3} x^{3}-28 c d \,e^{2} f \,g^{2} x^{3}+70 c \,e^{3} f^{2} g \,x^{3}+168 a d \,e^{2} g^{3} x^{2}-24 a \,e^{3} f \,g^{2} x^{2}+28 b \,d^{2} e \,g^{3} x^{2}-200 b d \,e^{2} f \,g^{2} x^{2}+28 b \,e^{3} f^{2} g \,x^{2}+21 c \,d^{3} g^{3} x^{2}-101 c \,d^{2} e f \,g^{2} x^{2}+259 c d \,e^{2} f^{2} g \,x^{2}-35 c \,e^{3} f^{3} x^{2}+210 a \,d^{2} e \,g^{3} x -84 a d \,e^{2} f \,g^{2} x +18 a \,e^{3} f^{2} g x +35 b \,d^{3} g^{3} x -259 b \,d^{2} e f \,g^{2} x +101 b d \,e^{2} f^{2} g x -21 b \,e^{3} f^{3} x -28 c \,d^{3} f \,g^{2} x +200 c \,d^{2} e \,f^{2} g x -28 c d \,e^{2} f^{3} x +105 a \,d^{3} g^{3}-105 a \,d^{2} e f \,g^{2}+63 a d \,e^{2} f^{2} g -15 a \,e^{3} f^{3}-70 b \,d^{3} f \,g^{2}+28 b \,d^{2} e \,f^{2} g -6 b d \,e^{2} f^{3}+56 c \,d^{3} f^{2} g -8 c \,d^{2} e \,f^{3}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (d g -e f \right )^{4}}\)	\(427\)
gosper	\(\frac {2 \sqrt {g x +f}\, \left (48 a \,e^{3} g^{3} x^{3}+8 b d \,e^{2} g^{3} x^{3}-56 b \,e^{3} f \,g^{2} x^{3}+6 c \,d^{2} e \,g^{3} x^{3}-28 c d \,e^{2} f \,g^{2} x^{3}+70 c \,e^{3} f^{2} g \,x^{3}+168 a d \,e^{2} g^{3} x^{2}-24 a \,e^{3} f \,g^{2} x^{2}+28 b \,d^{2} e \,g^{3} x^{2}-200 b d \,e^{2} f \,g^{2} x^{2}+28 b \,e^{3} f^{2} g \,x^{2}+21 c \,d^{3} g^{3} x^{2}-101 c \,d^{2} e f \,g^{2} x^{2}+259 c d \,e^{2} f^{2} g \,x^{2}-35 c \,e^{3} f^{3} x^{2}+210 a \,d^{2} e \,g^{3} x -84 a d \,e^{2} f \,g^{2} x +18 a \,e^{3} f^{2} g x +35 b \,d^{3} g^{3} x -259 b \,d^{2} e f \,g^{2} x +101 b d \,e^{2} f^{2} g x -21 b \,e^{3} f^{3} x -28 c \,d^{3} f \,g^{2} x +200 c \,d^{2} e \,f^{2} g x -28 c d \,e^{2} f^{3} x +105 a \,d^{3} g^{3}-105 a \,d^{2} e f \,g^{2}+63 a d \,e^{2} f^{2} g -15 a \,e^{3} f^{3}-70 b \,d^{3} f \,g^{2}+28 b \,d^{2} e \,f^{2} g -6 b d \,e^{2} f^{3}+56 c \,d^{3} f^{2} g -8 c \,d^{2} e \,f^{3}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (g^{4} d^{4}-4 d^{3} e f \,g^{3}+6 d^{2} e^{2} f^{2} g^{2}-4 d \,e^{3} f^{3} g +e^{4} f^{4}\right )}\)	\(468\)

[In]

int((c*x^2+b*x+a)/(e*x+d)^(9/2)/(g*x+f)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/105*(g*x+f)^(1/2)*(48*a*e^3*g^3*x^3+8*b*d*e^2*g^3*x^3-56*b*e^3*f*g^2*x^3+6*c*d^2*e*g^3*x^3-28*c*d*e^2*f*g^2*
x^3+70*c*e^3*f^2*g*x^3+168*a*d*e^2*g^3*x^2-24*a*e^3*f*g^2*x^2+28*b*d^2*e*g^3*x^2-200*b*d*e^2*f*g^2*x^2+28*b*e^
3*f^2*g*x^2+21*c*d^3*g^3*x^2-101*c*d^2*e*f*g^2*x^2+259*c*d*e^2*f^2*g*x^2-35*c*e^3*f^3*x^2+210*a*d^2*e*g^3*x-84
*a*d*e^2*f*g^2*x+18*a*e^3*f^2*g*x+35*b*d^3*g^3*x-259*b*d^2*e*f*g^2*x+101*b*d*e^2*f^2*g*x-21*b*e^3*f^3*x-28*c*d
^3*f*g^2*x+200*c*d^2*e*f^2*g*x-28*c*d*e^2*f^3*x+105*a*d^3*g^3-105*a*d^2*e*f*g^2+63*a*d*e^2*f^2*g-15*a*e^3*f^3-
70*b*d^3*f*g^2+28*b*d^2*e*f^2*g-6*b*d*e^2*f^3+56*c*d^3*f^2*g-8*c*d^2*e*f^3)/(e*x+d)^(7/2)/(d*g-e*f)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (257) = 514\).

Time = 22.89 (sec) , antiderivative size = 641, normalized size of antiderivative = 2.28 \[ \int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx=\frac {2 \, {\left (105 \, a d^{3} g^{3} - {\left (8 \, c d^{2} e + 6 \, b d e^{2} + 15 \, a e^{3}\right )} f^{3} + 7 \, {\left (8 \, c d^{3} + 4 \, b d^{2} e + 9 \, a d e^{2}\right )} f^{2} g - 35 \, {\left (2 \, b d^{3} + 3 \, a d^{2} e\right )} f g^{2} + 2 \, {\left (35 \, c e^{3} f^{2} g - 14 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f g^{2} + {\left (3 \, c d^{2} e + 4 \, b d e^{2} + 24 \, a e^{3}\right )} g^{3}\right )} x^{3} - {\left (35 \, c e^{3} f^{3} - 7 \, {\left (37 \, c d e^{2} + 4 \, b e^{3}\right )} f^{2} g + {\left (101 \, c d^{2} e + 200 \, b d e^{2} + 24 \, a e^{3}\right )} f g^{2} - 7 \, {\left (3 \, c d^{3} + 4 \, b d^{2} e + 24 \, a d e^{2}\right )} g^{3}\right )} x^{2} - {\left (7 \, {\left (4 \, c d e^{2} + 3 \, b e^{3}\right )} f^{3} - {\left (200 \, c d^{2} e + 101 \, b d e^{2} + 18 \, a e^{3}\right )} f^{2} g + 7 \, {\left (4 \, c d^{3} + 37 \, b d^{2} e + 12 \, a d e^{2}\right )} f g^{2} - 35 \, {\left (b d^{3} + 6 \, a d^{2} e\right )} g^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{105 \, {\left (d^{4} e^{4} f^{4} - 4 \, d^{5} e^{3} f^{3} g + 6 \, d^{6} e^{2} f^{2} g^{2} - 4 \, d^{7} e f g^{3} + d^{8} g^{4} + {\left (e^{8} f^{4} - 4 \, d e^{7} f^{3} g + 6 \, d^{2} e^{6} f^{2} g^{2} - 4 \, d^{3} e^{5} f g^{3} + d^{4} e^{4} g^{4}\right )} x^{4} + 4 \, {\left (d e^{7} f^{4} - 4 \, d^{2} e^{6} f^{3} g + 6 \, d^{3} e^{5} f^{2} g^{2} - 4 \, d^{4} e^{4} f g^{3} + d^{5} e^{3} g^{4}\right )} x^{3} + 6 \, {\left (d^{2} e^{6} f^{4} - 4 \, d^{3} e^{5} f^{3} g + 6 \, d^{4} e^{4} f^{2} g^{2} - 4 \, d^{5} e^{3} f g^{3} + d^{6} e^{2} g^{4}\right )} x^{2} + 4 \, {\left (d^{3} e^{5} f^{4} - 4 \, d^{4} e^{4} f^{3} g + 6 \, d^{5} e^{3} f^{2} g^{2} - 4 \, d^{6} e^{2} f g^{3} + d^{7} e g^{4}\right )} x\right )}} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(9/2)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

2/105*(105*a*d^3*g^3 - (8*c*d^2*e + 6*b*d*e^2 + 15*a*e^3)*f^3 + 7*(8*c*d^3 + 4*b*d^2*e + 9*a*d*e^2)*f^2*g - 35
*(2*b*d^3 + 3*a*d^2*e)*f*g^2 + 2*(35*c*e^3*f^2*g - 14*(c*d*e^2 + 2*b*e^3)*f*g^2 + (3*c*d^2*e + 4*b*d*e^2 + 24*
a*e^3)*g^3)*x^3 - (35*c*e^3*f^3 - 7*(37*c*d*e^2 + 4*b*e^3)*f^2*g + (101*c*d^2*e + 200*b*d*e^2 + 24*a*e^3)*f*g^
2 - 7*(3*c*d^3 + 4*b*d^2*e + 24*a*d*e^2)*g^3)*x^2 - (7*(4*c*d*e^2 + 3*b*e^3)*f^3 - (200*c*d^2*e + 101*b*d*e^2
+ 18*a*e^3)*f^2*g + 7*(4*c*d^3 + 37*b*d^2*e + 12*a*d*e^2)*f*g^2 - 35*(b*d^3 + 6*a*d^2*e)*g^3)*x)*sqrt(e*x + d)
*sqrt(g*x + f)/(d^4*e^4*f^4 - 4*d^5*e^3*f^3*g + 6*d^6*e^2*f^2*g^2 - 4*d^7*e*f*g^3 + d^8*g^4 + (e^8*f^4 - 4*d*e
^7*f^3*g + 6*d^2*e^6*f^2*g^2 - 4*d^3*e^5*f*g^3 + d^4*e^4*g^4)*x^4 + 4*(d*e^7*f^4 - 4*d^2*e^6*f^3*g + 6*d^3*e^5
*f^2*g^2 - 4*d^4*e^4*f*g^3 + d^5*e^3*g^4)*x^3 + 6*(d^2*e^6*f^4 - 4*d^3*e^5*f^3*g + 6*d^4*e^4*f^2*g^2 - 4*d^5*e
^3*f*g^3 + d^6*e^2*g^4)*x^2 + 4*(d^3*e^5*f^4 - 4*d^4*e^4*f^3*g + 6*d^5*e^3*f^2*g^2 - 4*d^6*e^2*f*g^3 + d^7*e*g
^4)*x)

Sympy [F]

\[ \int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx=\int \frac {a + b x + c x^{2}}{\left (d + e x\right )^{\frac {9}{2}} \sqrt {f + g x}}\, dx \]

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**(9/2)/(g*x+f)**(1/2),x)

[Out]

Integral((a + b*x + c*x**2)/((d + e*x)**(9/2)*sqrt(f + g*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(9/2)/(g*x+f)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e*(d*g-e*f)>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2035 vs. \(2 (257) = 514\).

Time = 0.52 (sec) , antiderivative size = 2035, normalized size of antiderivative = 7.24 \[ \int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(9/2)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

8/105*(35*sqrt(e*g)*c*e^10*f^5*g - 119*sqrt(e*g)*c*d*e^9*f^4*g^2 - 28*sqrt(e*g)*b*e^10*f^4*g^2 + 150*sqrt(e*g)
*c*d^2*e^8*f^3*g^3 + 88*sqrt(e*g)*b*d*e^9*f^3*g^3 + 24*sqrt(e*g)*a*e^10*f^3*g^3 - 86*sqrt(e*g)*c*d^3*e^7*f^2*g
^4 - 96*sqrt(e*g)*b*d^2*e^8*f^2*g^4 - 72*sqrt(e*g)*a*d*e^9*f^2*g^4 + 23*sqrt(e*g)*c*d^4*e^6*f*g^5 + 40*sqrt(e*
g)*b*d^3*e^7*f*g^5 + 72*sqrt(e*g)*a*d^2*e^8*f*g^5 - 3*sqrt(e*g)*c*d^5*e^5*g^6 - 4*sqrt(e*g)*b*d^4*e^6*g^6 - 24
*sqrt(e*g)*a*d^3*e^7*g^6 - 245*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*c*e
^8*f^4*g + 588*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*c*d*e^7*f^3*g^2 + 1
96*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*b*e^8*f^3*g^2 - 462*sqrt(e*g)*(
sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*c*d^2*e^6*f^2*g^3 - 420*sqrt(e*g)*(sqrt(e*g)*
sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*b*d*e^7*f^2*g^3 - 168*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d
) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*a*e^8*f^2*g^3 + 140*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f
+ (e*x + d)*e*g - d*e*g))^2*c*d^3*e^5*f*g^4 + 252*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)
*e*g - d*e*g))^2*b*d^2*e^6*f*g^4 + 336*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g
))^2*a*d*e^7*f*g^4 - 21*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*c*d^4*e^4*
g^5 - 28*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*b*d^3*e^5*g^5 - 168*sqrt(
e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*a*d^2*e^6*g^5 + 630*sqrt(e*g)*(sqrt(e*g
)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*c*e^6*f^3*g - 714*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d)
- sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*c*d*e^5*f^2*g^2 - 588*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f
+ (e*x + d)*e*g - d*e*g))^4*b*e^6*f^2*g^2 + 42*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*
g - d*e*g))^4*c*d^2*e^4*f*g^3 + 672*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^
4*b*d*e^5*f*g^3 + 504*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*a*e^6*f*g^3
+ 42*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*c*d^3*e^3*g^4 - 84*sqrt(e*g)*
(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*b*d^2*e^4*g^4 - 504*sqrt(e*g)*(sqrt(e*g)*sqr
t(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^4*a*d*e^5*g^4 - 770*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqr
t(e^2*f + (e*x + d)*e*g - d*e*g))^6*c*e^4*f^2*g + 140*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x +
d)*e*g - d*e*g))^6*c*d*e^3*f*g^2 + 700*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*
g))^6*b*e^4*f*g^2 - 210*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^6*c*d^2*e^2*
g^3 + 140*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^6*b*d*e^3*g^3 - 840*sqrt(e
*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^6*a*e^4*g^3 + 455*sqrt(e*g)*(sqrt(e*g)*sqr
t(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^8*c*e^2*f*g + 105*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(
e^2*f + (e*x + d)*e*g - d*e*g))^8*c*d*e*g^2 - 280*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*
e*g - d*e*g))^8*b*e^2*g^2 - 105*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^10*c
*g)/((e^2*f - d*e*g - (sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2)^7*abs(e))

Mupad [B] (verification not implemented)

Time = 13.50 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.61 \[ \int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx=\frac {\sqrt {f+g\,x}\,\left (\frac {x^3\,\left (12\,c\,d^2\,e\,g^3-56\,c\,d\,e^2\,f\,g^2+16\,b\,d\,e^2\,g^3+140\,c\,e^3\,f^2\,g-112\,b\,e^3\,f\,g^2+96\,a\,e^3\,g^3\right )}{105\,e^3\,{\left (d\,g-e\,f\right )}^4}-\frac {-112\,c\,d^3\,f^2\,g+140\,b\,d^3\,f\,g^2-210\,a\,d^3\,g^3+16\,c\,d^2\,e\,f^3-56\,b\,d^2\,e\,f^2\,g+210\,a\,d^2\,e\,f\,g^2+12\,b\,d\,e^2\,f^3-126\,a\,d\,e^2\,f^2\,g+30\,a\,e^3\,f^3}{105\,e^3\,{\left (d\,g-e\,f\right )}^4}+\frac {x\,\left (-56\,c\,d^3\,f\,g^2+70\,b\,d^3\,g^3+400\,c\,d^2\,e\,f^2\,g-518\,b\,d^2\,e\,f\,g^2+420\,a\,d^2\,e\,g^3-56\,c\,d\,e^2\,f^3+202\,b\,d\,e^2\,f^2\,g-168\,a\,d\,e^2\,f\,g^2-42\,b\,e^3\,f^3+36\,a\,e^3\,f^2\,g\right )}{105\,e^3\,{\left (d\,g-e\,f\right )}^4}+\frac {2\,x^2\,\left (7\,d\,g-e\,f\right )\,\left (3\,c\,d^2\,g^2-14\,c\,d\,e\,f\,g+4\,b\,d\,e\,g^2+35\,c\,e^2\,f^2-28\,b\,e^2\,f\,g+24\,a\,e^2\,g^2\right )}{105\,e^3\,{\left (d\,g-e\,f\right )}^4}\right )}{x^3\,\sqrt {d+e\,x}+\frac {d^3\,\sqrt {d+e\,x}}{e^3}+\frac {3\,d\,x^2\,\sqrt {d+e\,x}}{e}+\frac {3\,d^2\,x\,\sqrt {d+e\,x}}{e^2}} \]

[In]

int((a + b*x + c*x^2)/((f + g*x)^(1/2)*(d + e*x)^(9/2)),x)

[Out]

((f + g*x)^(1/2)*((x^3*(96*a*e^3*g^3 + 16*b*d*e^2*g^3 + 12*c*d^2*e*g^3 - 112*b*e^3*f*g^2 + 140*c*e^3*f^2*g - 5
6*c*d*e^2*f*g^2))/(105*e^3*(d*g - e*f)^4) - (30*a*e^3*f^3 - 210*a*d^3*g^3 + 12*b*d*e^2*f^3 + 16*c*d^2*e*f^3 +
140*b*d^3*f*g^2 - 112*c*d^3*f^2*g - 126*a*d*e^2*f^2*g + 210*a*d^2*e*f*g^2 - 56*b*d^2*e*f^2*g)/(105*e^3*(d*g -
e*f)^4) + (x*(70*b*d^3*g^3 - 42*b*e^3*f^3 + 420*a*d^2*e*g^3 - 56*c*d*e^2*f^3 + 36*a*e^3*f^2*g - 56*c*d^3*f*g^2
- 168*a*d*e^2*f*g^2 + 202*b*d*e^2*f^2*g - 518*b*d^2*e*f*g^2 + 400*c*d^2*e*f^2*g))/(105*e^3*(d*g - e*f)^4) + (
2*x^2*(7*d*g - e*f)*(24*a*e^2*g^2 + 3*c*d^2*g^2 + 35*c*e^2*f^2 + 4*b*d*e*g^2 - 28*b*e^2*f*g - 14*c*d*e*f*g))/(
105*e^3*(d*g - e*f)^4)))/(x^3*(d + e*x)^(1/2) + (d^3*(d + e*x)^(1/2))/e^3 + (3*d*x^2*(d + e*x)^(1/2))/e + (3*d
^2*x*(d + e*x)^(1/2))/e^2)