3.8.19 \(\int \frac {\sqrt {d+e x}}{(f+g x)^{9/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [719]

Optimal. Leaf size=267 \[ \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

2/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^(7/2)/(e*x+d)^(1/2)+12/35*c*d*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^(5/2)/(e*x+d)^(1/2)+16/35*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^3/(g*x+f)^(3/2)/(e*x+d)^(1/2)+32/35*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)/(-a*e*g+c*d*f)^4/(e*x+d)^(1/2)/(g*x+f)^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {886, 874} \begin {gather*} \frac {32 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^4}+\frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac {12 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 \sqrt {d+e x} (f+g x)^{7/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^(7/2)) + (12*c*d*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(5/2)) + (16*c^2*d^2*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (32*c^3*d^
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*Sqrt[f + g*x])

Rule 874

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))),
x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d
*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]

Rule 886

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

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x}}{(f+g x)^{9/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {(6 c d) \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 (c d f-a e g)}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {\left (24 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 (c d f-a e g)^2}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {\left (16 c^3 d^3\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 (c d f-a e g)^3}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 152, normalized size = 0.57 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-5 a^3 e^3 g^3+3 a^2 c d e^2 g^2 (7 f+2 g x)-a c^2 d^2 e g \left (35 f^2+28 f g x+8 g^2 x^2\right )+c^3 d^3 \left (35 f^3+70 f^2 g x+56 f g^2 x^2+16 g^3 x^3\right )\right )}{35 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-5*a^3*e^3*g^3 + 3*a^2*c*d*e^2*g^2*(7*f + 2*g*x) - a*c^2*d^2*e*g*(35*f^2 + 2
8*f*g*x + 8*g^2*x^2) + c^3*d^3*(35*f^3 + 70*f^2*g*x + 56*f*g^2*x^2 + 16*g^3*x^3)))/(35*(c*d*f - a*e*g)^4*Sqrt[
d + e*x]*(f + g*x)^(7/2))

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Maple [A]
time = 0.14, size = 183, normalized size = 0.69

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}-56 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x +28 a \,c^{2} d^{2} e f \,g^{2} x -70 c^{3} d^{3} f^{2} g x +5 a^{3} e^{3} g^{3}-21 a^{2} c d \,e^{2} f \,g^{2}+35 a \,c^{2} d^{2} e \,f^{2} g -35 f^{3} c^{3} d^{3}\right )}{35 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right )^{4}}\)	\(183\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}-56 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x +28 a \,c^{2} d^{2} e f \,g^{2} x -70 c^{3} d^{3} f^{2} g x +5 a^{3} e^{3} g^{3}-21 a^{2} c d \,e^{2} f \,g^{2}+35 a \,c^{2} d^{2} e \,f^{2} g -35 f^{3} c^{3} d^{3}\right ) \sqrt {e x +d}}{35 \left (g x +f \right )^{\frac {7}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35/(e*x+d)^(1/2)/(g*x+f)^(7/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(-16*c^3*d^3*g^3*x^3+8*a*c^2*d^2*e*g^3*x^2-56*c^
3*d^3*f*g^2*x^2-6*a^2*c*d*e^2*g^3*x+28*a*c^2*d^2*e*f*g^2*x-70*c^3*d^3*f^2*g*x+5*a^3*e^3*g^3-21*a^2*c*d*e^2*f*g
^2+35*a*c^2*d^2*e*f^2*g-35*c^3*d^3*f^3)/(a*e*g-c*d*f)^4

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1002 vs. \(2 (247) = 494\).
time = 4.10, size = 1002, normalized size = 3.75 \begin {gather*} \frac {2 \, {\left (16 \, c^{3} d^{3} g^{3} x^{3} + 56 \, c^{3} d^{3} f g^{2} x^{2} + 70 \, c^{3} d^{3} f^{2} g x + 35 \, c^{3} d^{3} f^{3} - 5 \, a^{3} g^{3} e^{3} + 3 \, {\left (2 \, a^{2} c d g^{3} x + 7 \, a^{2} c d f g^{2}\right )} e^{2} - {\left (8 \, a c^{2} d^{2} g^{3} x^{2} + 28 \, a c^{2} d^{2} f g^{2} x + 35 \, a c^{2} d^{2} f^{2} g\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{35 \, {\left (c^{4} d^{5} f^{4} g^{4} x^{4} + 4 \, c^{4} d^{5} f^{5} g^{3} x^{3} + 6 \, c^{4} d^{5} f^{6} g^{2} x^{2} + 4 \, c^{4} d^{5} f^{7} g x + c^{4} d^{5} f^{8} + {\left (a^{4} g^{8} x^{5} + 4 \, a^{4} f g^{7} x^{4} + 6 \, a^{4} f^{2} g^{6} x^{3} + 4 \, a^{4} f^{3} g^{5} x^{2} + a^{4} f^{4} g^{4} x\right )} e^{5} - {\left (4 \, a^{3} c d f g^{7} x^{5} - a^{4} d f^{4} g^{4} + {\left (16 \, a^{3} c d f^{2} g^{6} - a^{4} d g^{8}\right )} x^{4} + 4 \, {\left (6 \, a^{3} c d f^{3} g^{5} - a^{4} d f g^{7}\right )} x^{3} + 2 \, {\left (8 \, a^{3} c d f^{4} g^{4} - 3 \, a^{4} d f^{2} g^{6}\right )} x^{2} + 4 \, {\left (a^{3} c d f^{5} g^{3} - a^{4} d f^{3} g^{5}\right )} x\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{6} x^{5} - 2 \, a^{3} c d^{2} f^{5} g^{3} + 2 \, {\left (6 \, a^{2} c^{2} d^{2} f^{3} g^{5} - a^{3} c d^{2} f g^{7}\right )} x^{4} + 2 \, {\left (9 \, a^{2} c^{2} d^{2} f^{4} g^{4} - 4 \, a^{3} c d^{2} f^{2} g^{6}\right )} x^{3} + 12 \, {\left (a^{2} c^{2} d^{2} f^{5} g^{3} - a^{3} c d^{2} f^{3} g^{5}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{2} f^{6} g^{2} - 8 \, a^{3} c d^{2} f^{4} g^{4}\right )} x\right )} e^{3} - 2 \, {\left (2 \, a c^{3} d^{3} f^{3} g^{5} x^{5} - 3 \, a^{2} c^{2} d^{3} f^{6} g^{2} + {\left (8 \, a c^{3} d^{3} f^{4} g^{4} - 3 \, a^{2} c^{2} d^{3} f^{2} g^{6}\right )} x^{4} + 12 \, {\left (a c^{3} d^{3} f^{5} g^{3} - a^{2} c^{2} d^{3} f^{3} g^{5}\right )} x^{3} + 2 \, {\left (4 \, a c^{3} d^{3} f^{6} g^{2} - 9 \, a^{2} c^{2} d^{3} f^{4} g^{4}\right )} x^{2} + 2 \, {\left (a c^{3} d^{3} f^{7} g - 6 \, a^{2} c^{2} d^{3} f^{5} g^{3}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{4} g^{4} x^{5} - 4 \, a c^{3} d^{4} f^{7} g + 4 \, {\left (c^{4} d^{4} f^{5} g^{3} - a c^{3} d^{4} f^{3} g^{5}\right )} x^{4} + 2 \, {\left (3 \, c^{4} d^{4} f^{6} g^{2} - 8 \, a c^{3} d^{4} f^{4} g^{4}\right )} x^{3} + 4 \, {\left (c^{4} d^{4} f^{7} g - 6 \, a c^{3} d^{4} f^{5} g^{3}\right )} x^{2} + {\left (c^{4} d^{4} f^{8} - 16 \, a c^{3} d^{4} f^{6} g^{2}\right )} x\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(16*c^3*d^3*g^3*x^3 + 56*c^3*d^3*f*g^2*x^2 + 70*c^3*d^3*f^2*g*x + 35*c^3*d^3*f^3 - 5*a^3*g^3*e^3 + 3*(2*a
^2*c*d*g^3*x + 7*a^2*c*d*f*g^2)*e^2 - (8*a*c^2*d^2*g^3*x^2 + 28*a*c^2*d^2*f*g^2*x + 35*a*c^2*d^2*f^2*g)*e)*sqr
t(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g*x + f)*sqrt(x*e + d)/(c^4*d^5*f^4*g^4*x^4 + 4*c^4*d^5*f^5*g^3*
x^3 + 6*c^4*d^5*f^6*g^2*x^2 + 4*c^4*d^5*f^7*g*x + c^4*d^5*f^8 + (a^4*g^8*x^5 + 4*a^4*f*g^7*x^4 + 6*a^4*f^2*g^6
*x^3 + 4*a^4*f^3*g^5*x^2 + a^4*f^4*g^4*x)*e^5 - (4*a^3*c*d*f*g^7*x^5 - a^4*d*f^4*g^4 + (16*a^3*c*d*f^2*g^6 - a
^4*d*g^8)*x^4 + 4*(6*a^3*c*d*f^3*g^5 - a^4*d*f*g^7)*x^3 + 2*(8*a^3*c*d*f^4*g^4 - 3*a^4*d*f^2*g^6)*x^2 + 4*(a^3
*c*d*f^5*g^3 - a^4*d*f^3*g^5)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^6*x^5 - 2*a^3*c*d^2*f^5*g^3 + 2*(6*a^2*c^2*d^2*f
^3*g^5 - a^3*c*d^2*f*g^7)*x^4 + 2*(9*a^2*c^2*d^2*f^4*g^4 - 4*a^3*c*d^2*f^2*g^6)*x^3 + 12*(a^2*c^2*d^2*f^5*g^3
- a^3*c*d^2*f^3*g^5)*x^2 + (3*a^2*c^2*d^2*f^6*g^2 - 8*a^3*c*d^2*f^4*g^4)*x)*e^3 - 2*(2*a*c^3*d^3*f^3*g^5*x^5 -
3*a^2*c^2*d^3*f^6*g^2 + (8*a*c^3*d^3*f^4*g^4 - 3*a^2*c^2*d^3*f^2*g^6)*x^4 + 12*(a*c^3*d^3*f^5*g^3 - a^2*c^2*d
^3*f^3*g^5)*x^3 + 2*(4*a*c^3*d^3*f^6*g^2 - 9*a^2*c^2*d^3*f^4*g^4)*x^2 + 2*(a*c^3*d^3*f^7*g - 6*a^2*c^2*d^3*f^5
*g^3)*x)*e^2 + (c^4*d^4*f^4*g^4*x^5 - 4*a*c^3*d^4*f^7*g + 4*(c^4*d^4*f^5*g^3 - a*c^3*d^4*f^3*g^5)*x^4 + 2*(3*c
^4*d^4*f^6*g^2 - 8*a*c^3*d^4*f^4*g^4)*x^3 + 4*(c^4*d^4*f^7*g - 6*a*c^3*d^4*f^5*g^3)*x^2 + (c^4*d^4*f^8 - 16*a*
c^3*d^4*f^6*g^2)*x)*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((e*x+d)**(1/2)/(g*x+f)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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

[Out]

Timed out

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Mupad [B]
time = 5.51, size = 357, normalized size = 1.34 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (10\,a^3\,e^3\,g^3-42\,a^2\,c\,d\,e^2\,f\,g^2+70\,a\,c^2\,d^2\,e\,f^2\,g-70\,c^3\,d^3\,f^3\right )}{35\,e\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^3\,d^3\,x^3\,\sqrt {d+e\,x}}{35\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c\,d\,x\,\sqrt {d+e\,x}\,\left (3\,a^2\,e^2\,g^2-14\,a\,c\,d\,e\,f\,g+35\,c^2\,d^2\,f^2\right )}{35\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^2\,d^2\,x^2\,\left (a\,e\,g-7\,c\,d\,f\right )\,\sqrt {d+e\,x}}{35\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {d\,f^3\,\sqrt {f+g\,x}}{e\,g^3}+\frac {x^3\,\sqrt {f+g\,x}\,\left (d\,g+3\,e\,f\right )}{e\,g}+\frac {3\,f\,x^2\,\sqrt {f+g\,x}\,\left (d\,g+e\,f\right )}{e\,g^2}+\frac {f^2\,x\,\sqrt {f+g\,x}\,\left (3\,d\,g+e\,f\right )}{e\,g^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(((d + e*x)^(1/2)*(10*a^3*e^3*g^3 - 70*c^3*d^3*f^3 + 70*a*c^2*
d^2*e*f^2*g - 42*a^2*c*d*e^2*f*g^2))/(35*e*g^3*(a*e*g - c*d*f)^4) - (32*c^3*d^3*x^3*(d + e*x)^(1/2))/(35*e*(a*
e*g - c*d*f)^4) - (4*c*d*x*(d + e*x)^(1/2)*(3*a^2*e^2*g^2 + 35*c^2*d^2*f^2 - 14*a*c*d*e*f*g))/(35*e*g^2*(a*e*g
- c*d*f)^4) + (16*c^2*d^2*x^2*(a*e*g - 7*c*d*f)*(d + e*x)^(1/2))/(35*e*g*(a*e*g - c*d*f)^4)))/(x^4*(f + g*x)^
(1/2) + (d*f^3*(f + g*x)^(1/2))/(e*g^3) + (x^3*(f + g*x)^(1/2)*(d*g + 3*e*f))/(e*g) + (3*f*x^2*(f + g*x)^(1/2)
*(d*g + e*f))/(e*g^2) + (f^2*x*(f + g*x)^(1/2)*(3*d*g + e*f))/(e*g^3))

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