\(\int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx\) [840]

Optimal result

Integrand size = 29, antiderivative size = 198 \[ \int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{5 (e f-d g) (d+e x)^{5/2}}+\frac {2 (2 c d (5 e f-3 d g)-e (5 b e f-b d g-4 a e g)) \sqrt {f+g x}}{15 e^2 (e f-d g)^2 (d+e x)^{3/2}}+\frac {2 \left (2 e g (5 b e f-b d g-4 a e g)-c \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{15 e^2 (e f-d g)^3 \sqrt {d+e x}} \]

[Out]

-2/5*(a+d*(-b*e+c*d)/e^2)*(g*x+f)^(1/2)/(-d*g+e*f)/(e*x+d)^(5/2)+2/15*(2*c*d*(-3*d*g+5*e*f)-e*(-4*a*e*g-b*d*g+
5*b*e*f))*(g*x+f)^(1/2)/e^2/(-d*g+e*f)^2/(e*x+d)^(3/2)+2/15*(2*e*g*(-4*a*e*g-b*d*g+5*b*e*f)-c*(3*d^2*g^2-10*d*
e*f*g+15*e^2*f^2))*(g*x+f)^(1/2)/e^2/(-d*g+e*f)^3/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {963, 79, 37} \[ \int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (2 e g (-4 a e g-b d g+5 b e f)-c \left (3 d^2 g^2-10 d e f g+15 e^2 f^2\right )\right )}{15 e^2 \sqrt {d+e x} (e f-d g)^3}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{5 (d+e x)^{5/2} (e f-d g)}+\frac {2 \sqrt {f+g x} (2 c d (5 e f-3 d g)-e (-4 a e g-b d g+5 b e f))}{15 e^2 (d+e x)^{3/2} (e f-d g)^2} \]

[In]

Int[(a + b*x + c*x^2)/((d + e*x)^(7/2)*Sqrt[f + g*x]),x]

[Out]

(-2*(a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/(5*(e*f - d*g)*(d + e*x)^(5/2)) + (2*(2*c*d*(5*e*f - 3*d*g) - e*(
5*b*e*f - b*d*g - 4*a*e*g))*Sqrt[f + g*x])/(15*e^2*(e*f - d*g)^2*(d + e*x)^(3/2)) + (2*(2*e*g*(5*b*e*f - b*d*g
- 4*a*e*g) - c*(15*e^2*f^2 - 10*d*e*f*g + 3*d^2*g^2))*Sqrt[f + g*x])/(15*e^2*(e*f - d*g)^3*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 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}}{5 (e f-d g) (d+e x)^{5/2}}-\frac {2 \int \frac {\frac {c d (5 e f-d g)-e (5 b e f-b d g-4 a e g)}{2 e^2}-\frac {5}{2} c \left (f-\frac {d g}{e}\right ) x}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx}{5 (e f-d g)} \\ & = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{5 (e f-d g) (d+e x)^{5/2}}+\frac {2 (2 c d (5 e f-3 d g)-e (5 b e f-b d g-4 a e g)) \sqrt {f+g x}}{15 e^2 (e f-d g)^2 (d+e x)^{3/2}}-\frac {\left (2 e g (5 b e f-b d g-4 a e g)-c \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right )\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx}{15 e^2 (e f-d g)^2} \\ & = -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{5 (e f-d g) (d+e x)^{5/2}}+\frac {2 (2 c d (5 e f-3 d g)-e (5 b e f-b d g-4 a e g)) \sqrt {f+g x}}{15 e^2 (e f-d g)^2 (d+e x)^{3/2}}+\frac {2 \left (2 e g (5 b e f-b d g-4 a e g)-c \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{15 e^2 (e f-d g)^3 \sqrt {d+e x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx=-\frac {2 \sqrt {f+g x} \left (15 c f^2-15 b f g+15 a g^2-\frac {10 c d f (f+g x)}{d+e x}+\frac {5 b e f (f+g x)}{d+e x}+\frac {5 b d g (f+g x)}{d+e x}-\frac {10 a e g (f+g x)}{d+e x}+\frac {3 c d^2 (f+g x)^2}{(d+e x)^2}-\frac {3 b d e (f+g x)^2}{(d+e x)^2}+\frac {3 a e^2 (f+g x)^2}{(d+e x)^2}\right )}{15 (e f-d g)^3 \sqrt {d+e x}} \]

[In]

Integrate[(a + b*x + c*x^2)/((d + e*x)^(7/2)*Sqrt[f + g*x]),x]

[Out]

(-2*Sqrt[f + g*x]*(15*c*f^2 - 15*b*f*g + 15*a*g^2 - (10*c*d*f*(f + g*x))/(d + e*x) + (5*b*e*f*(f + g*x))/(d +
e*x) + (5*b*d*g*(f + g*x))/(d + e*x) - (10*a*e*g*(f + g*x))/(d + e*x) + (3*c*d^2*(f + g*x)^2)/(d + e*x)^2 - (3
*b*d*e*(f + g*x)^2)/(d + e*x)^2 + (3*a*e^2*(f + g*x)^2)/(d + e*x)^2))/(15*(e*f - d*g)^3*Sqrt[d + e*x])

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {2 \sqrt {g x +f}\, \left (8 a \,e^{2} g^{2} x^{2}+2 b d e \,g^{2} x^{2}-10 b \,e^{2} f g \,x^{2}+3 c \,d^{2} g^{2} x^{2}-10 c d e f g \,x^{2}+15 c \,e^{2} f^{2} x^{2}+20 a d e \,g^{2} x -4 a \,e^{2} f g x +5 b \,d^{2} g^{2} x -26 b d e f g x +5 b \,e^{2} f^{2} x -4 c \,d^{2} f g x +20 c d e \,f^{2} x +15 a \,d^{2} g^{2}-10 a d e f g +3 a \,e^{2} f^{2}-10 b \,d^{2} f g +2 b d e \,f^{2}+8 c \,d^{2} f^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (d g -e f \right )^{3}}\)	\(210\)
gosper	\(\frac {2 \sqrt {g x +f}\, \left (8 a \,e^{2} g^{2} x^{2}+2 b d e \,g^{2} x^{2}-10 b \,e^{2} f g \,x^{2}+3 c \,d^{2} g^{2} x^{2}-10 c d e f g \,x^{2}+15 c \,e^{2} f^{2} x^{2}+20 a d e \,g^{2} x -4 a \,e^{2} f g x +5 b \,d^{2} g^{2} x -26 b d e f g x +5 b \,e^{2} f^{2} x -4 c \,d^{2} f g x +20 c d e \,f^{2} x +15 a \,d^{2} g^{2}-10 a d e f g +3 a \,e^{2} f^{2}-10 b \,d^{2} f g +2 b d e \,f^{2}+8 c \,d^{2} f^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (d^{3} g^{3}-3 d^{2} e f \,g^{2}+3 d \,e^{2} f^{2} g -e^{3} f^{3}\right )}\)	\(238\)

[In]

int((c*x^2+b*x+a)/(e*x+d)^(7/2)/(g*x+f)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/15*(g*x+f)^(1/2)*(8*a*e^2*g^2*x^2+2*b*d*e*g^2*x^2-10*b*e^2*f*g*x^2+3*c*d^2*g^2*x^2-10*c*d*e*f*g*x^2+15*c*e^2
*f^2*x^2+20*a*d*e*g^2*x-4*a*e^2*f*g*x+5*b*d^2*g^2*x-26*b*d*e*f*g*x+5*b*e^2*f^2*x-4*c*d^2*f*g*x+20*c*d*e*f^2*x+
15*a*d^2*g^2-10*a*d*e*f*g+3*a*e^2*f^2-10*b*d^2*f*g+2*b*d*e*f^2+8*c*d^2*f^2)/(e*x+d)^(5/2)/(d*g-e*f)^3

Fricas [A] (verification not implemented)

none

Time = 7.21 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.78 \[ \int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx=-\frac {2 \, {\left (15 \, a d^{2} g^{2} + {\left (8 \, c d^{2} + 2 \, b d e + 3 \, a e^{2}\right )} f^{2} - 10 \, {\left (b d^{2} + a d e\right )} f g + {\left (15 \, c e^{2} f^{2} - 10 \, {\left (c d e + b e^{2}\right )} f g + {\left (3 \, c d^{2} + 2 \, b d e + 8 \, a e^{2}\right )} g^{2}\right )} x^{2} + {\left (5 \, {\left (4 \, c d e + b e^{2}\right )} f^{2} - 2 \, {\left (2 \, c d^{2} + 13 \, b d e + 2 \, a e^{2}\right )} f g + 5 \, {\left (b d^{2} + 4 \, a d e\right )} g^{2}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{15 \, {\left (d^{3} e^{3} f^{3} - 3 \, d^{4} e^{2} f^{2} g + 3 \, d^{5} e f g^{2} - d^{6} g^{3} + {\left (e^{6} f^{3} - 3 \, d e^{5} f^{2} g + 3 \, d^{2} e^{4} f g^{2} - d^{3} e^{3} g^{3}\right )} x^{3} + 3 \, {\left (d e^{5} f^{3} - 3 \, d^{2} e^{4} f^{2} g + 3 \, d^{3} e^{3} f g^{2} - d^{4} e^{2} g^{3}\right )} x^{2} + 3 \, {\left (d^{2} e^{4} f^{3} - 3 \, d^{3} e^{3} f^{2} g + 3 \, d^{4} e^{2} f g^{2} - d^{5} e g^{3}\right )} x\right )}} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(7/2)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

-2/15*(15*a*d^2*g^2 + (8*c*d^2 + 2*b*d*e + 3*a*e^2)*f^2 - 10*(b*d^2 + a*d*e)*f*g + (15*c*e^2*f^2 - 10*(c*d*e +
b*e^2)*f*g + (3*c*d^2 + 2*b*d*e + 8*a*e^2)*g^2)*x^2 + (5*(4*c*d*e + b*e^2)*f^2 - 2*(2*c*d^2 + 13*b*d*e + 2*a*
e^2)*f*g + 5*(b*d^2 + 4*a*d*e)*g^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)/(d^3*e^3*f^3 - 3*d^4*e^2*f^2*g + 3*d^5*e*f*
g^2 - d^6*g^3 + (e^6*f^3 - 3*d*e^5*f^2*g + 3*d^2*e^4*f*g^2 - d^3*e^3*g^3)*x^3 + 3*(d*e^5*f^3 - 3*d^2*e^4*f^2*g
+ 3*d^3*e^3*f*g^2 - d^4*e^2*g^3)*x^2 + 3*(d^2*e^4*f^3 - 3*d^3*e^3*f^2*g + 3*d^4*e^2*f*g^2 - d^5*e*g^3)*x)

Sympy [F]

\[ \int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx=\int \frac {a + b x + c x^{2}}{\left (d + e x\right )^{\frac {7}{2}} \sqrt {f + g x}}\, dx \]

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**(7/2)/(g*x+f)**(1/2),x)

[Out]

Integral((a + b*x + c*x**2)/((d + e*x)**(7/2)*sqrt(f + g*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(7/2)/(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. 1175 vs. \(2 (180) = 360\).

Time = 0.42 (sec) , antiderivative size = 1175, normalized size of antiderivative = 5.93 \[ \int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx=-\frac {4 \, {\left (15 \, \sqrt {e g} c e^{8} f^{4} - 40 \, \sqrt {e g} c d e^{7} f^{3} g - 10 \, \sqrt {e g} b e^{8} f^{3} g + 38 \, \sqrt {e g} c d^{2} e^{6} f^{2} g^{2} + 22 \, \sqrt {e g} b d e^{7} f^{2} g^{2} + 8 \, \sqrt {e g} a e^{8} f^{2} g^{2} - 16 \, \sqrt {e g} c d^{3} e^{5} f g^{3} - 14 \, \sqrt {e g} b d^{2} e^{6} f g^{3} - 16 \, \sqrt {e g} a d e^{7} f g^{3} + 3 \, \sqrt {e g} c d^{4} e^{4} g^{4} + 2 \, \sqrt {e g} b d^{3} e^{5} g^{4} + 8 \, \sqrt {e g} a d^{2} e^{6} g^{4} - 60 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} c e^{6} f^{3} + 80 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} c d e^{5} f^{2} g + 50 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} b e^{6} f^{2} g - 20 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} c d^{2} e^{4} f g^{2} - 60 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} b d e^{5} f g^{2} - 40 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} a e^{6} f g^{2} + 10 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} b d^{2} e^{4} g^{3} + 40 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2} a d e^{5} g^{3} + 90 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} c e^{4} f^{2} - 40 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} c d e^{3} f g - 70 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} b e^{4} f g + 30 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} c d^{2} e^{2} g^{2} - 10 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} b d e^{3} g^{2} + 80 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{4} a e^{4} g^{2} - 60 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{6} c e^{2} f + 30 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{6} b e^{2} g + 15 \, \sqrt {e g} {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{8} c\right )}}{15 \, {\left (e^{2} f - d e g - {\left (\sqrt {e g} \sqrt {e x + d} - \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g}\right )}^{2}\right )}^{5} e {\left | e \right |}} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(7/2)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

-4/15*(15*sqrt(e*g)*c*e^8*f^4 - 40*sqrt(e*g)*c*d*e^7*f^3*g - 10*sqrt(e*g)*b*e^8*f^3*g + 38*sqrt(e*g)*c*d^2*e^6
*f^2*g^2 + 22*sqrt(e*g)*b*d*e^7*f^2*g^2 + 8*sqrt(e*g)*a*e^8*f^2*g^2 - 16*sqrt(e*g)*c*d^3*e^5*f*g^3 - 14*sqrt(e
*g)*b*d^2*e^6*f*g^3 - 16*sqrt(e*g)*a*d*e^7*f*g^3 + 3*sqrt(e*g)*c*d^4*e^4*g^4 + 2*sqrt(e*g)*b*d^3*e^5*g^4 + 8*s
qrt(e*g)*a*d^2*e^6*g^4 - 60*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^6*
f^3 + 80*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^5*f^2*g + 50*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^6*f^2*g - 20*sqrt(e*g)*(sqrt(e*g)*sq
rt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^2*c*d^2*e^4*f*g^2 - 60*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^5*f*g^2 - 40*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^6*f*g^2 + 10*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^4*g^3 + 40*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^5
*g^3 + 90*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^4*f^2 - 40*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^3*f*g - 70*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^4*f*g + 30*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^2*g^2 - 10*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^3*g^2 + 80*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^4*g^2 - 60*sqrt(e*g)*(sqrt(e*g)*sqrt(e*x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^6*c*e^2*f + 30*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^2*g + 15*sqrt(e*g)*(sqrt(e*g)*sqrt(e*
x + d) - sqrt(e^2*f + (e*x + d)*e*g - d*e*g))^8*c)/((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)^5*e*abs(e))

Mupad [B] (verification not implemented)

Time = 13.21 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.31 \[ \int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx=\frac {\sqrt {f+g\,x}\,\left (\frac {16\,c\,d^2\,f^2-20\,b\,d^2\,f\,g+30\,a\,d^2\,g^2+4\,b\,d\,e\,f^2-20\,a\,d\,e\,f\,g+6\,a\,e^2\,f^2}{15\,e^2\,{\left (d\,g-e\,f\right )}^3}+\frac {x\,\left (-8\,c\,d^2\,f\,g+10\,b\,d^2\,g^2+40\,c\,d\,e\,f^2-52\,b\,d\,e\,f\,g+40\,a\,d\,e\,g^2+10\,b\,e^2\,f^2-8\,a\,e^2\,f\,g\right )}{15\,e^2\,{\left (d\,g-e\,f\right )}^3}+\frac {x^2\,\left (6\,c\,d^2\,g^2-20\,c\,d\,e\,f\,g+4\,b\,d\,e\,g^2+30\,c\,e^2\,f^2-20\,b\,e^2\,f\,g+16\,a\,e^2\,g^2\right )}{15\,e^2\,{\left (d\,g-e\,f\right )}^3}\right )}{x^2\,\sqrt {d+e\,x}+\frac {d^2\,\sqrt {d+e\,x}}{e^2}+\frac {2\,d\,x\,\sqrt {d+e\,x}}{e}} \]

[In]

int((a + b*x + c*x^2)/((f + g*x)^(1/2)*(d + e*x)^(7/2)),x)

[Out]

((f + g*x)^(1/2)*((30*a*d^2*g^2 + 6*a*e^2*f^2 + 16*c*d^2*f^2 + 4*b*d*e*f^2 - 20*b*d^2*f*g - 20*a*d*e*f*g)/(15*
e^2*(d*g - e*f)^3) + (x*(10*b*d^2*g^2 + 10*b*e^2*f^2 + 40*a*d*e*g^2 + 40*c*d*e*f^2 - 8*a*e^2*f*g - 8*c*d^2*f*g
- 52*b*d*e*f*g))/(15*e^2*(d*g - e*f)^3) + (x^2*(16*a*e^2*g^2 + 6*c*d^2*g^2 + 30*c*e^2*f^2 + 4*b*d*e*g^2 - 20*
b*e^2*f*g - 20*c*d*e*f*g))/(15*e^2*(d*g - e*f)^3)))/(x^2*(d + e*x)^(1/2) + (d^2*(d + e*x)^(1/2))/e^2 + (2*d*x*
(d + e*x)^(1/2))/e)