3.8.2 \(\int \frac {(f+g x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [702]

Optimal. Leaf size=200 \[ -\frac {8 (c d f-a e g) \left (2 a e^2 g-c d (9 e f-7 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 c^3 d^3 e (d+e x)^{7/2}}+\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}} \]

[Out]

-8/693*(-a*e*g+c*d*f)*(2*a*e^2*g-c*d*(-7*d*g+9*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3/e/(e*x+d)
^(7/2)+8/99*g*(-a*e*g+c*d*f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2/e/(e*x+d)^(5/2)+2/11*(g*x+f)^2*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/(e*x+d)^(7/2)

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Rubi [A]
time = 0.16, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {884, 808, 662} \begin {gather*} -\frac {8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g) \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{693 c^3 d^3 e (d+e x)^{7/2}}+\frac {8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(-8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(693*c^3*
d^3*e*(d + e*x)^(7/2)) + (8*g*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(99*c^2*d^2*e*(d
+ e*x)^(5/2)) + (2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*c*d*(d + e*x)^(7/2))

Rule 662

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

Rule 808

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

Rule 884

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

Rubi steps

\begin {align*} \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}}+\frac {(4 (c d f-a e g)) \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{11 c d}\\ &=\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}}+\frac {\left (4 (c d f-a e g) \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{99 c d}\\ &=\frac {8 (c d f-a e g) \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 c^2 d^2 (d+e x)^{7/2}}+\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 100, normalized size = 0.50 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (8 a^2 e^2 g^2-4 a c d e g (11 f+7 g x)+c^2 d^2 \left (99 f^2+154 f g x+63 g^2 x^2\right )\right )}{693 c^3 d^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(11*f + 7*g*x) + c^2*d^2*(99*f^2
+ 154*f*g*x + 63*g^2*x^2)))/(693*c^3*d^3*Sqrt[d + e*x])

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Maple [A]
time = 0.13, size = 108, normalized size = 0.54

method	result	size
default	\(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{3} \left (63 g^{2} x^{2} c^{2} d^{2}-28 a c d e \,g^{2} x +154 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-44 a c d e f g +99 f^{2} c^{2} d^{2}\right )}{693 \sqrt {e x +d}\, c^{3} d^{3}}\)	\(108\)
gosper	\(\frac {2 \left (c d x +a e \right ) \left (63 g^{2} x^{2} c^{2} d^{2}-28 a c d e \,g^{2} x +154 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-44 a c d e f g +99 f^{2} c^{2} d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/693*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^3*(63*c^2*d^2*g^2*x^2-28*a*c*d*e*g^2*x+154*c^2*d^2
*f*g*x+8*a^2*e^2*g^2-44*a*c*d*e*f*g+99*c^2*d^2*f^2)/c^3/d^3

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Maxima [A]
time = 0.35, size = 240, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} x^{2} e + 3 \, a^{2} c d x e^{2} + a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{2}}{7 \, c d} + \frac {4 \, {\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} x^{3} e + 15 \, a^{2} c^{2} d^{2} x^{2} e^{2} + a^{3} c d x e^{3} - 2 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f g}{63 \, c^{2} d^{2}} + \frac {2 \, {\left (63 \, c^{5} d^{5} x^{5} + 161 \, a c^{4} d^{4} x^{4} e + 113 \, a^{2} c^{3} d^{3} x^{3} e^{2} + 3 \, a^{3} c^{2} d^{2} x^{2} e^{3} - 4 \, a^{4} c d x e^{4} + 8 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} g^{2}}{693 \, c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*x^2*e + 3*a^2*c*d*x*e^2 + a^3*e^3)*sqrt(c*d*x + a*e)*f^2/(c*d) + 4/63*(7*c^4*d^
4*x^4 + 19*a*c^3*d^3*x^3*e + 15*a^2*c^2*d^2*x^2*e^2 + a^3*c*d*x*e^3 - 2*a^4*e^4)*sqrt(c*d*x + a*e)*f*g/(c^2*d^
2) + 2/693*(63*c^5*d^5*x^5 + 161*a*c^4*d^4*x^4*e + 113*a^2*c^3*d^3*x^3*e^2 + 3*a^3*c^2*d^2*x^2*e^3 - 4*a^4*c*d
*x*e^4 + 8*a^5*e^5)*sqrt(c*d*x + a*e)*g^2/(c^3*d^3)

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Fricas [A]
time = 1.28, size = 282, normalized size = 1.41 \begin {gather*} \frac {2 \, {\left (63 \, c^{5} d^{5} g^{2} x^{5} + 154 \, c^{5} d^{5} f g x^{4} + 99 \, c^{5} d^{5} f^{2} x^{3} + 8 \, a^{5} g^{2} e^{5} - 4 \, {\left (a^{4} c d g^{2} x + 11 \, a^{4} c d f g\right )} e^{4} + {\left (3 \, a^{3} c^{2} d^{2} g^{2} x^{2} + 22 \, a^{3} c^{2} d^{2} f g x + 99 \, a^{3} c^{2} d^{2} f^{2}\right )} e^{3} + {\left (113 \, a^{2} c^{3} d^{3} g^{2} x^{3} + 330 \, a^{2} c^{3} d^{3} f g x^{2} + 297 \, a^{2} c^{3} d^{3} f^{2} x\right )} e^{2} + {\left (161 \, a c^{4} d^{4} g^{2} x^{4} + 418 \, a c^{4} d^{4} f g x^{3} + 297 \, a c^{4} d^{4} f^{2} x^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{693 \, {\left (c^{3} d^{3} x e + c^{3} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/693*(63*c^5*d^5*g^2*x^5 + 154*c^5*d^5*f*g*x^4 + 99*c^5*d^5*f^2*x^3 + 8*a^5*g^2*e^5 - 4*(a^4*c*d*g^2*x + 11*a
^4*c*d*f*g)*e^4 + (3*a^3*c^2*d^2*g^2*x^2 + 22*a^3*c^2*d^2*f*g*x + 99*a^3*c^2*d^2*f^2)*e^3 + (113*a^2*c^3*d^3*g
^2*x^3 + 330*a^2*c^3*d^3*f*g*x^2 + 297*a^2*c^3*d^3*f^2*x)*e^2 + (161*a*c^4*d^4*g^2*x^4 + 418*a*c^4*d^4*f*g*x^3
+ 297*a*c^4*d^4*f^2*x^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c^3*d^3*x*e + c^3*d^4)

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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((g*x+f)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1976 vs. \(2 (187) = 374\).
time = 4.92, size = 1976, normalized size = 9.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/3465*(33*c^2*d^2*f^2*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-
c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*c*d*e - c
*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*d*e - c*d
^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^(-1) - 22*c^2*d^2*f*g*((35*sqrt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c
*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*
e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3
*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a
*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4))*e^(-1) + c^2*d^2*g^2*((315*sqrt(-c*d^2*
e + a*e^3)*c^5*d^10 - 35*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2 - 40*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^6*e^4 - 48
*sqrt(-c*d^2*e + a*e^3)*a^3*c^2*d^4*e^6 - 64*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8 - 128*sqrt(-c*d^2*e + a*e^3)
*a^5*e^10)*e^(-4)/(c^5*d^5) + (1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((x*e + d)*c*d*e
- c*d^2*e + a*e^3)^(5/2)*a^3*e^9 + 2970*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((x*e + d)*c
*d*e - c*d^2*e + a*e^3)^(9/2)*a*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))*e^(-9)/(c^5*d^5))*e^(-1)
- 462*a*c*d*f^2*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(
5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^
2*e + a*e^3)*a^2*e^4)/(c^2*d^2))*e^(-1) + 132*a*c*d*f*g*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e
+ a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^
3*d^3) + (35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*
a*e^3 + 15*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3)) - 22*a*c*d*g^2*((35*sqrt(-c*d^2*e + a*
e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt(-c*d
^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c*d*e - c*
d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)*c*d*e -
c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4)) + 1155*a^2*f^2*
(((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*e^(-1)/(c*d) + (sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^2*e + a*e^
3)*a*e^2)/(c*d))*e + 33*a^2*g^2*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 -
4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*
c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*
d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e - 462*a^2*f*g*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*
a*e^3 - 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sq
rt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2)))*e^(-1)

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Mupad [B]
time = 3.56, size = 259, normalized size = 1.30 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,a^5\,e^5\,g^2-88\,a^4\,c\,d\,e^4\,f\,g+198\,a^3\,c^2\,d^2\,e^3\,f^2}{693\,c^3\,d^3}+\frac {x^3\,\left (226\,a^2\,c^3\,d^3\,e^2\,g^2+836\,a\,c^4\,d^4\,e\,f\,g+198\,c^5\,d^5\,f^2\right )}{693\,c^3\,d^3}+\frac {2\,c^2\,d^2\,g^2\,x^5}{11}+\frac {2\,c\,d\,g\,x^4\,\left (23\,a\,e\,g+22\,c\,d\,f\right )}{99}+\frac {2\,a^2\,e^2\,x\,\left (-4\,a^2\,e^2\,g^2+22\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,c^2\,d^2}+\frac {2\,a\,e\,x^2\,\left (a^2\,e^2\,g^2+110\,a\,c\,d\,e\,f\,g+99\,c^2\,d^2\,f^2\right )}{231\,c\,d}\right )}{\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x)

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((16*a^5*e^5*g^2 + 198*a^3*c^2*d^2*e^3*f^2 - 88*a^4*c*d*e^4*f*g
)/(693*c^3*d^3) + (x^3*(198*c^5*d^5*f^2 + 226*a^2*c^3*d^3*e^2*g^2 + 836*a*c^4*d^4*e*f*g))/(693*c^3*d^3) + (2*c
^2*d^2*g^2*x^5)/11 + (2*c*d*g*x^4*(23*a*e*g + 22*c*d*f))/99 + (2*a^2*e^2*x*(297*c^2*d^2*f^2 - 4*a^2*e^2*g^2 +
22*a*c*d*e*f*g))/(693*c^2*d^2) + (2*a*e*x^2*(a^2*e^2*g^2 + 99*c^2*d^2*f^2 + 110*a*c*d*e*f*g))/(231*c*d)))/(d +
e*x)^(1/2)

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