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

Optimal. Leaf size=281 \[ -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac {2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt {f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}+\frac {2 \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^3 (d+e x)^{3/2}}-\frac {4 g \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^4 \sqrt {d+e x}} \]

[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)

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Rubi [A]
time = 0.18, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {963, 79, 47, 37} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 \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*}

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Mathematica [A]
time = 0.35, size = 301, normalized size = 1.07 \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[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))

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Maple [A]
time = 0.08, size = 427, normalized size = 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	\(\frac {2 \sqrt {g x +f}\, \left (48 a \,e^{3} g^{3} x^{3}+8 b d \,e^{2} g^{3} x^{3}-56 b \,e^{3} f \,g^{2} x^{3}+6 c \,d^{2} e \,g^{3} x^{3}-28 c d \,e^{2} f \,g^{2} x^{3}+70 c \,e^{3} f^{2} g \,x^{3}+168 a d \,e^{2} g^{3} x^{2}-24 a \,e^{3} f \,g^{2} x^{2}+28 b \,d^{2} e \,g^{3} x^{2}-200 b d \,e^{2} f \,g^{2} x^{2}+28 b \,e^{3} f^{2} g \,x^{2}+21 c \,d^{3} g^{3} x^{2}-101 c \,d^{2} e f \,g^{2} x^{2}+259 c d \,e^{2} f^{2} g \,x^{2}-35 c \,e^{3} f^{3} x^{2}+210 a \,d^{2} e \,g^{3} x -84 a d \,e^{2} f \,g^{2} x +18 a \,e^{3} f^{2} g x +35 b \,d^{3} g^{3} x -259 b \,d^{2} e f \,g^{2} x +101 b d \,e^{2} f^{2} g x -21 b \,e^{3} f^{3} x -28 c \,d^{3} f \,g^{2} x +200 c \,d^{2} e \,f^{2} g x -28 c d \,e^{2} f^{3} x +105 a \,d^{3} g^{3}-105 a \,d^{2} e f \,g^{2}+63 a d \,e^{2} f^{2} g -15 a \,e^{3} f^{3}-70 b \,d^{3} f \,g^{2}+28 b \,d^{2} e \,f^{2} g -6 b d \,e^{2} f^{3}+56 c \,d^{3} f^{2} g -8 c \,d^{2} e \,f^{3}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (g^{4} d^{4}-4 e \,g^{3} f \,d^{3}+6 d^{2} e^{2} f^{2} g^{2}-4 d \,e^{3} f^{3} g +e^{4} f^{4}\right )}\)	\(468\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(d*g-%e*f>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (267) = 534\).
time = 84.76, size = 658, normalized size = 2.34 \begin {gather*} \frac {2 \, {\left (21 \, c d^{3} g^{3} x^{2} + 56 \, c d^{3} f^{2} g - 70 \, b d^{3} f g^{2} + 105 \, a d^{3} g^{3} - 7 \, {\left (4 \, c d^{3} f g^{2} - 5 \, b d^{3} g^{3}\right )} x - {\left (15 \, a f^{3} - 2 \, {\left (35 \, c f^{2} g - 28 \, b f g^{2} + 24 \, a g^{3}\right )} x^{3} + {\left (35 \, c f^{3} - 28 \, b f^{2} g + 24 \, a f g^{2}\right )} x^{2} + 3 \, {\left (7 \, b f^{3} - 6 \, a f^{2} g\right )} x\right )} e^{3} - {\left (6 \, b d f^{3} - 63 \, a d f^{2} g + 4 \, {\left (7 \, c d f g^{2} - 2 \, b d g^{3}\right )} x^{3} - {\left (259 \, c d f^{2} g - 200 \, b d f g^{2} + 168 \, a d g^{3}\right )} x^{2} + {\left (28 \, c d f^{3} - 101 \, b d f^{2} g + 84 \, a d f g^{2}\right )} x\right )} e^{2} + {\left (6 \, c d^{2} g^{3} x^{3} - 8 \, c d^{2} f^{3} + 28 \, b d^{2} f^{2} g - 105 \, a d^{2} f g^{2} - {\left (101 \, c d^{2} f g^{2} - 28 \, b d^{2} g^{3}\right )} x^{2} + {\left (200 \, c d^{2} f^{2} g - 259 \, b d^{2} f g^{2} + 210 \, a d^{2} g^{3}\right )} x\right )} e\right )} \sqrt {g x + f} \sqrt {x e + d}}{105 \, {\left (d^{8} g^{4} + f^{4} x^{4} e^{8} - 4 \, {\left (d f^{3} g x^{4} - d f^{4} x^{3}\right )} e^{7} + 2 \, {\left (3 \, d^{2} f^{2} g^{2} x^{4} - 8 \, d^{2} f^{3} g x^{3} + 3 \, d^{2} f^{4} x^{2}\right )} e^{6} - 4 \, {\left (d^{3} f g^{3} x^{4} - 6 \, d^{3} f^{2} g^{2} x^{3} + 6 \, d^{3} f^{3} g x^{2} - d^{3} f^{4} x\right )} e^{5} + {\left (d^{4} g^{4} x^{4} - 16 \, d^{4} f g^{3} x^{3} + 36 \, d^{4} f^{2} g^{2} x^{2} - 16 \, d^{4} f^{3} g x + d^{4} f^{4}\right )} e^{4} + 4 \, {\left (d^{5} g^{4} x^{3} - 6 \, d^{5} f g^{3} x^{2} + 6 \, d^{5} f^{2} g^{2} x - d^{5} f^{3} g\right )} e^{3} + 2 \, {\left (3 \, d^{6} g^{4} x^{2} - 8 \, d^{6} f g^{3} x + 3 \, d^{6} f^{2} g^{2}\right )} e^{2} + 4 \, {\left (d^{7} g^{4} x - d^{7} f g^{3}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(21*c*d^3*g^3*x^2 + 56*c*d^3*f^2*g - 70*b*d^3*f*g^2 + 105*a*d^3*g^3 - 7*(4*c*d^3*f*g^2 - 5*b*d^3*g^3)*x
- (15*a*f^3 - 2*(35*c*f^2*g - 28*b*f*g^2 + 24*a*g^3)*x^3 + (35*c*f^3 - 28*b*f^2*g + 24*a*f*g^2)*x^2 + 3*(7*b*f
^3 - 6*a*f^2*g)*x)*e^3 - (6*b*d*f^3 - 63*a*d*f^2*g + 4*(7*c*d*f*g^2 - 2*b*d*g^3)*x^3 - (259*c*d*f^2*g - 200*b*
d*f*g^2 + 168*a*d*g^3)*x^2 + (28*c*d*f^3 - 101*b*d*f^2*g + 84*a*d*f*g^2)*x)*e^2 + (6*c*d^2*g^3*x^3 - 8*c*d^2*f
^3 + 28*b*d^2*f^2*g - 105*a*d^2*f*g^2 - (101*c*d^2*f*g^2 - 28*b*d^2*g^3)*x^2 + (200*c*d^2*f^2*g - 259*b*d^2*f*
g^2 + 210*a*d^2*g^3)*x)*e)*sqrt(g*x + f)*sqrt(x*e + d)/(d^8*g^4 + f^4*x^4*e^8 - 4*(d*f^3*g*x^4 - d*f^4*x^3)*e^
7 + 2*(3*d^2*f^2*g^2*x^4 - 8*d^2*f^3*g*x^3 + 3*d^2*f^4*x^2)*e^6 - 4*(d^3*f*g^3*x^4 - 6*d^3*f^2*g^2*x^3 + 6*d^3
*f^3*g*x^2 - d^3*f^4*x)*e^5 + (d^4*g^4*x^4 - 16*d^4*f*g^3*x^3 + 36*d^4*f^2*g^2*x^2 - 16*d^4*f^3*g*x + d^4*f^4)
*e^4 + 4*(d^5*g^4*x^3 - 6*d^5*f*g^3*x^2 + 6*d^5*f^2*g^2*x - d^5*f^3*g)*e^3 + 2*(3*d^6*g^4*x^2 - 8*d^6*f*g^3*x
+ 3*d^6*f^2*g^2)*e^2 + 4*(d^7*g^4*x - d^7*f*g^3)*e)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 752 vs. \(2 (267) = 534\).
time = 2.51, size = 752, normalized size = 2.68 \begin {gather*} \frac {2 \, {\left ({\left ({\left (g x + f\right )} {\left (\frac {2 \, {\left (3 \, c d^{2} g^{10} e^{4} - 14 \, c d f g^{9} e^{5} + 4 \, b d g^{10} e^{5} + 35 \, c f^{2} g^{8} e^{6} - 28 \, b f g^{9} e^{6} + 24 \, a g^{10} e^{6}\right )} {\left (g x + f\right )}}{d^{4} g^{6} {\left | g \right |} e^{3} - 4 \, d^{3} f g^{5} {\left | g \right |} e^{4} + 6 \, d^{2} f^{2} g^{4} {\left | g \right |} e^{5} - 4 \, d f^{3} g^{3} {\left | g \right |} e^{6} + f^{4} g^{2} {\left | g \right |} e^{7}} + \frac {7 \, {\left (3 \, c d^{3} g^{11} e^{3} - 17 \, c d^{2} f g^{10} e^{4} + 4 \, b d^{2} g^{11} e^{4} + 49 \, c d f^{2} g^{9} e^{5} - 32 \, b d f g^{10} e^{5} + 24 \, a d g^{11} e^{5} - 35 \, c f^{3} g^{8} e^{6} + 28 \, b f^{2} g^{9} e^{6} - 24 \, a f g^{10} e^{6}\right )}}{d^{4} g^{6} {\left | g \right |} e^{3} - 4 \, d^{3} f g^{5} {\left | g \right |} e^{4} + 6 \, d^{2} f^{2} g^{4} {\left | g \right |} e^{5} - 4 \, d f^{3} g^{3} {\left | g \right |} e^{6} + f^{4} g^{2} {\left | g \right |} e^{7}}\right )} - \frac {35 \, {\left (2 \, c d^{3} f g^{11} e^{3} - b d^{3} g^{12} e^{3} - 12 \, c d^{2} f^{2} g^{10} e^{4} + 9 \, b d^{2} f g^{11} e^{4} - 6 \, a d^{2} g^{12} e^{4} + 18 \, c d f^{3} g^{9} e^{5} - 15 \, b d f^{2} g^{10} e^{5} + 12 \, a d f g^{11} e^{5} - 8 \, c f^{4} g^{8} e^{6} + 7 \, b f^{3} g^{9} e^{6} - 6 \, a f^{2} g^{10} e^{6}\right )}}{d^{4} g^{6} {\left | g \right |} e^{3} - 4 \, d^{3} f g^{5} {\left | g \right |} e^{4} + 6 \, d^{2} f^{2} g^{4} {\left | g \right |} e^{5} - 4 \, d f^{3} g^{3} {\left | g \right |} e^{6} + f^{4} g^{2} {\left | g \right |} e^{7}}\right )} {\left (g x + f\right )} + \frac {105 \, {\left (c d^{3} f^{2} g^{11} e^{3} - b d^{3} f g^{12} e^{3} + a d^{3} g^{13} e^{3} - 3 \, c d^{2} f^{3} g^{10} e^{4} + 3 \, b d^{2} f^{2} g^{11} e^{4} - 3 \, a d^{2} f g^{12} e^{4} + 3 \, c d f^{4} g^{9} e^{5} - 3 \, b d f^{3} g^{10} e^{5} + 3 \, a d f^{2} g^{11} e^{5} - c f^{5} g^{8} e^{6} + b f^{4} g^{9} e^{6} - a f^{3} g^{10} e^{6}\right )}}{d^{4} g^{6} {\left | g \right |} e^{3} - 4 \, d^{3} f g^{5} {\left | g \right |} e^{4} + 6 \, d^{2} f^{2} g^{4} {\left | g \right |} e^{5} - 4 \, d f^{3} g^{3} {\left | g \right |} e^{6} + f^{4} g^{2} {\left | g \right |} e^{7}}\right )} \sqrt {g x + f}}{105 \, {\left (d g^{2} + {\left (g x + f\right )} g e - f g e\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(((g*x + f)*(2*(3*c*d^2*g^10*e^4 - 14*c*d*f*g^9*e^5 + 4*b*d*g^10*e^5 + 35*c*f^2*g^8*e^6 - 28*b*f*g^9*e^6
+ 24*a*g^10*e^6)*(g*x + f)/(d^4*g^6*abs(g)*e^3 - 4*d^3*f*g^5*abs(g)*e^4 + 6*d^2*f^2*g^4*abs(g)*e^5 - 4*d*f^3*
g^3*abs(g)*e^6 + f^4*g^2*abs(g)*e^7) + 7*(3*c*d^3*g^11*e^3 - 17*c*d^2*f*g^10*e^4 + 4*b*d^2*g^11*e^4 + 49*c*d*f
^2*g^9*e^5 - 32*b*d*f*g^10*e^5 + 24*a*d*g^11*e^5 - 35*c*f^3*g^8*e^6 + 28*b*f^2*g^9*e^6 - 24*a*f*g^10*e^6)/(d^4
*g^6*abs(g)*e^3 - 4*d^3*f*g^5*abs(g)*e^4 + 6*d^2*f^2*g^4*abs(g)*e^5 - 4*d*f^3*g^3*abs(g)*e^6 + f^4*g^2*abs(g)*
e^7)) - 35*(2*c*d^3*f*g^11*e^3 - b*d^3*g^12*e^3 - 12*c*d^2*f^2*g^10*e^4 + 9*b*d^2*f*g^11*e^4 - 6*a*d^2*g^12*e^
4 + 18*c*d*f^3*g^9*e^5 - 15*b*d*f^2*g^10*e^5 + 12*a*d*f*g^11*e^5 - 8*c*f^4*g^8*e^6 + 7*b*f^3*g^9*e^6 - 6*a*f^2
*g^10*e^6)/(d^4*g^6*abs(g)*e^3 - 4*d^3*f*g^5*abs(g)*e^4 + 6*d^2*f^2*g^4*abs(g)*e^5 - 4*d*f^3*g^3*abs(g)*e^6 +
f^4*g^2*abs(g)*e^7))*(g*x + f) + 105*(c*d^3*f^2*g^11*e^3 - b*d^3*f*g^12*e^3 + a*d^3*g^13*e^3 - 3*c*d^2*f^3*g^1
0*e^4 + 3*b*d^2*f^2*g^11*e^4 - 3*a*d^2*f*g^12*e^4 + 3*c*d*f^4*g^9*e^5 - 3*b*d*f^3*g^10*e^5 + 3*a*d*f^2*g^11*e^
5 - c*f^5*g^8*e^6 + b*f^4*g^9*e^6 - a*f^3*g^10*e^6)/(d^4*g^6*abs(g)*e^3 - 4*d^3*f*g^5*abs(g)*e^4 + 6*d^2*f^2*g
^4*abs(g)*e^5 - 4*d*f^3*g^3*abs(g)*e^6 + f^4*g^2*abs(g)*e^7))*sqrt(g*x + f)/(d*g^2 + (g*x + f)*g*e - f*g*e)^(7
/2)

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Mupad [B]
time = 4.65, size = 452, normalized size = 1.61 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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