3.783 \(\int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=501 \[ \frac {128 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{3465 c^6 d^6 e g \sqrt {d+e x}}-\frac {128 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )}{3465 c^5 d^5 e}-\frac {32 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (10 a e^2 g+c d (e f-11 d g)\right )}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}} \]

[Out]

128/3465*(-a*e*g+c*d*f)^3*(10*a*e^2*g+c*d*(-11*d*g+e*f))*(2*a*e^2*g-c*d*(-d*g+3*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)/c^6/d^6/e/g/(e*x+d)^(1/2)-32/1155*(-a*e*g+c*d*f)^2*(10*a*e^2*g+c*d*(-11*d*g+e*f))*(g*x+f)^2*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/g/(e*x+d)^(1/2)-16/693*(-a*e*g+c*d*f)*(10*a*e^2*g+c*d*(-11*d*g+e
*f))*(g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/g/(e*x+d)^(1/2)-2/99*(10*a*e^2*g+c*d*(-11*d*g+e
*f))*(g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/g/(e*x+d)^(1/2)+2/11*e*(g*x+f)^5*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/g/(e*x+d)^(1/2)-128/3465*(-a*e*g+c*d*f)^3*(10*a*e^2*g+c*d*(-11*d*g+e*f))*(e*x+d)
^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^5/d^5/e

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Rubi [A]  time = 0.89, antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {880, 870, 794, 648} \[ -\frac {2 (f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (10 a e^2 g+c d (e f-11 d g)\right )}{99 c^2 d^2 g \sqrt {d+e x}}-\frac {16 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {32 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {128 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )}{3465 c^5 d^5 e}+\frac {128 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{3465 c^6 d^6 e g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(3/2)*(f + g*x)^4)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(2*a*e^2*g - c*d*(3*e*f - d*g))*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(3465*c^6*d^6*e*g*Sqrt[d + e*x]) - (128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*
d*g))*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3465*c^5*d^5*e) - (32*(c*d*f - a*e*g)^2*(10*
a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(1155*c^4*d^4*g*Sqrt[d
+ e*x]) - (16*(c*d*f - a*e*g)*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2])/(693*c^3*d^3*g*Sqrt[d + e*x]) - (2*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^4*Sqrt[a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2])/(99*c^2*d^2*g*Sqrt[d + e*x]) + (2*e*(f + g*x)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2])/(11*c*d*g*Sqrt[d + e*x])

Rule 648

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 794

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 870

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] &&  !Integ
erQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

Rule 880

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 - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/(c*g*(n + p + 2)), x] - Dist[(b*e*g*(
n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(c*g*(n + p + 2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x +
c*x^2)^p, x], 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] && Eq
Q[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p - 1, 0] &&  !LtQ[n, -1] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {1}{11} \left (-11 d+\frac {10 a e^2}{c d}+\frac {e f}{g}\right ) \int \frac {\sqrt {d+e x} (f+g x)^4}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {\left (8 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{99 c^2 d^2 g}\\ &=-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {\left (16 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{231 c^3 d^3 g}\\ &=-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}-\frac {\left (64 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1155 c^4 d^4 g}\\ &=-\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^5 d^5 e}-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}+\frac {\left (64 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3465 c^5 d^5 e g}\\ &=\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^6 d^6 e g \sqrt {d+e x}}-\frac {128 (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3465 c^5 d^5 e}-\frac {32 (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1155 c^4 d^4 g \sqrt {d+e x}}-\frac {16 (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{693 c^3 d^3 g \sqrt {d+e x}}-\frac {2 \left (10 a e^2 g+c d (e f-11 d g)\right ) (f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{99 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{11 c d g \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 246, normalized size = 0.49 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (3465 \left (c d^2-a e^2\right ) (c d f-a e g)^4-385 g^3 (a e+c d x)^4 \left (5 a e^2 g-c d (d g+4 e f)\right )+990 g^2 (a e+c d x)^3 (c d f-a e g) \left (c d (2 d g+3 e f)-5 a e^2 g\right )+1386 g (a e+c d x)^2 (c d f-a e g)^2 \left (c d (3 d g+2 e f)-5 a e^2 g\right )+1155 (a e+c d x) (c d f-a e g)^3 \left (c d (4 d g+e f)-5 a e^2 g\right )+315 e g^4 (a e+c d x)^5\right )}{3465 c^6 d^6 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(3/2)*(f + g*x)^4)/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)]*(3465*(c*d^2 - a*e^2)*(c*d*f - a*e*g)^4 + 1155*(c*d*f - a*e*g)^3*(-5*a*e^2*g
+ c*d*(e*f + 4*d*g))*(a*e + c*d*x) + 1386*g*(c*d*f - a*e*g)^2*(-5*a*e^2*g + c*d*(2*e*f + 3*d*g))*(a*e + c*d*x)
^2 + 990*g^2*(c*d*f - a*e*g)*(-5*a*e^2*g + c*d*(3*e*f + 2*d*g))*(a*e + c*d*x)^3 - 385*g^3*(5*a*e^2*g - c*d*(4*
e*f + d*g))*(a*e + c*d*x)^4 + 315*e*g^4*(a*e + c*d*x)^5))/(3465*c^6*d^6*Sqrt[d + e*x])

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fricas [A]  time = 0.67, size = 597, normalized size = 1.19 \[ \frac {2 \, {\left (315 \, c^{5} d^{5} e g^{4} x^{5} + 1155 \, {\left (3 \, c^{5} d^{6} - 2 \, a c^{4} d^{4} e^{2}\right )} f^{4} - 1848 \, {\left (5 \, a c^{4} d^{5} e - 4 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{3} g + 1584 \, {\left (7 \, a^{2} c^{3} d^{4} e^{2} - 6 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{2} g^{2} - 704 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} f g^{3} + 128 \, {\left (11 \, a^{4} c d^{2} e^{4} - 10 \, a^{5} e^{6}\right )} g^{4} + 35 \, {\left (44 \, c^{5} d^{5} e f g^{3} + {\left (11 \, c^{5} d^{6} - 10 \, a c^{4} d^{4} e^{2}\right )} g^{4}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{5} e f^{2} g^{2} + 22 \, {\left (9 \, c^{5} d^{6} - 8 \, a c^{4} d^{4} e^{2}\right )} f g^{3} - 4 \, {\left (11 \, a c^{4} d^{5} e - 10 \, a^{2} c^{3} d^{3} e^{3}\right )} g^{4}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{5} e f^{3} g + 99 \, {\left (7 \, c^{5} d^{6} - 6 \, a c^{4} d^{4} e^{2}\right )} f^{2} g^{2} - 44 \, {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} f g^{3} + 8 \, {\left (11 \, a^{2} c^{3} d^{4} e^{2} - 10 \, a^{3} c^{2} d^{2} e^{4}\right )} g^{4}\right )} x^{2} + {\left (1155 \, c^{5} d^{5} e f^{4} + 924 \, {\left (5 \, c^{5} d^{6} - 4 \, a c^{4} d^{4} e^{2}\right )} f^{3} g - 792 \, {\left (7 \, a c^{4} d^{5} e - 6 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{2} g^{2} + 352 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} f g^{3} - 64 \, {\left (11 \, a^{3} c^{2} d^{3} e^{3} - 10 \, a^{4} c d e^{5}\right )} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3465 \, {\left (c^{6} d^{6} e x + c^{6} d^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*c^5*d^5*e*g^4*x^5 + 1155*(3*c^5*d^6 - 2*a*c^4*d^4*e^2)*f^4 - 1848*(5*a*c^4*d^5*e - 4*a^2*c^3*d^3*e
^3)*f^3*g + 1584*(7*a^2*c^3*d^4*e^2 - 6*a^3*c^2*d^2*e^4)*f^2*g^2 - 704*(9*a^3*c^2*d^3*e^3 - 8*a^4*c*d*e^5)*f*g
^3 + 128*(11*a^4*c*d^2*e^4 - 10*a^5*e^6)*g^4 + 35*(44*c^5*d^5*e*f*g^3 + (11*c^5*d^6 - 10*a*c^4*d^4*e^2)*g^4)*x
^4 + 10*(297*c^5*d^5*e*f^2*g^2 + 22*(9*c^5*d^6 - 8*a*c^4*d^4*e^2)*f*g^3 - 4*(11*a*c^4*d^5*e - 10*a^2*c^3*d^3*e
^3)*g^4)*x^3 + 6*(462*c^5*d^5*e*f^3*g + 99*(7*c^5*d^6 - 6*a*c^4*d^4*e^2)*f^2*g^2 - 44*(9*a*c^4*d^5*e - 8*a^2*c
^3*d^3*e^3)*f*g^3 + 8*(11*a^2*c^3*d^4*e^2 - 10*a^3*c^2*d^2*e^4)*g^4)*x^2 + (1155*c^5*d^5*e*f^4 + 924*(5*c^5*d^
6 - 4*a*c^4*d^4*e^2)*f^3*g - 792*(7*a*c^4*d^5*e - 6*a^2*c^3*d^3*e^3)*f^2*g^2 + 352*(9*a^2*c^3*d^4*e^2 - 8*a^3*
c^2*d^2*e^4)*f*g^3 - 64*(11*a^3*c^2*d^3*e^3 - 10*a^4*c*d*e^5)*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)
*x)*sqrt(e*x + d)/(c^6*d^6*e*x + c^6*d^7)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{4}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(3/2)*(g*x + f)^4/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x), x)

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maple [A]  time = 0.01, size = 641, normalized size = 1.28 \[ -\frac {2 \left (c d x +a e \right ) \left (-315 e \,g^{4} x^{5} c^{5} d^{5}+350 a \,c^{4} d^{4} e^{2} g^{4} x^{4}-385 c^{5} d^{6} g^{4} x^{4}-1540 c^{5} d^{5} e f \,g^{3} x^{4}-400 a^{2} c^{3} d^{3} e^{3} g^{4} x^{3}+440 a \,c^{4} d^{5} e \,g^{4} x^{3}+1760 a \,c^{4} d^{4} e^{2} f \,g^{3} x^{3}-1980 c^{5} d^{6} f \,g^{3} x^{3}-2970 c^{5} d^{5} e \,f^{2} g^{2} x^{3}+480 a^{3} c^{2} d^{2} e^{4} g^{4} x^{2}-528 a^{2} c^{3} d^{4} e^{2} g^{4} x^{2}-2112 a^{2} c^{3} d^{3} e^{3} f \,g^{3} x^{2}+2376 a \,c^{4} d^{5} e f \,g^{3} x^{2}+3564 a \,c^{4} d^{4} e^{2} f^{2} g^{2} x^{2}-4158 c^{5} d^{6} f^{2} g^{2} x^{2}-2772 c^{5} d^{5} e \,f^{3} g \,x^{2}-640 a^{4} c d \,e^{5} g^{4} x +704 a^{3} c^{2} d^{3} e^{3} g^{4} x +2816 a^{3} c^{2} d^{2} e^{4} f \,g^{3} x -3168 a^{2} c^{3} d^{4} e^{2} f \,g^{3} x -4752 a^{2} c^{3} d^{3} e^{3} f^{2} g^{2} x +5544 a \,c^{4} d^{5} e \,f^{2} g^{2} x +3696 a \,c^{4} d^{4} e^{2} f^{3} g x -4620 c^{5} d^{6} f^{3} g x -1155 c^{5} d^{5} e \,f^{4} x +1280 a^{5} e^{6} g^{4}-1408 a^{4} c \,d^{2} e^{4} g^{4}-5632 a^{4} c d \,e^{5} f \,g^{3}+6336 a^{3} c^{2} d^{3} e^{3} f \,g^{3}+9504 a^{3} c^{2} d^{2} e^{4} f^{2} g^{2}-11088 a^{2} c^{3} d^{4} e^{2} f^{2} g^{2}-7392 a^{2} c^{3} d^{3} e^{3} f^{3} g +9240 a \,c^{4} d^{5} e \,f^{3} g +2310 a \,c^{4} d^{4} e^{2} f^{4}-3465 d^{6} f^{4} c^{5}\right ) \sqrt {e x +d}}{3465 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c^{6} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(g*x+f)^4/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2),x)

[Out]

-2/3465*(c*d*x+a*e)*(-315*c^5*d^5*e*g^4*x^5+350*a*c^4*d^4*e^2*g^4*x^4-385*c^5*d^6*g^4*x^4-1540*c^5*d^5*e*f*g^3
*x^4-400*a^2*c^3*d^3*e^3*g^4*x^3+440*a*c^4*d^5*e*g^4*x^3+1760*a*c^4*d^4*e^2*f*g^3*x^3-1980*c^5*d^6*f*g^3*x^3-2
970*c^5*d^5*e*f^2*g^2*x^3+480*a^3*c^2*d^2*e^4*g^4*x^2-528*a^2*c^3*d^4*e^2*g^4*x^2-2112*a^2*c^3*d^3*e^3*f*g^3*x
^2+2376*a*c^4*d^5*e*f*g^3*x^2+3564*a*c^4*d^4*e^2*f^2*g^2*x^2-4158*c^5*d^6*f^2*g^2*x^2-2772*c^5*d^5*e*f^3*g*x^2
-640*a^4*c*d*e^5*g^4*x+704*a^3*c^2*d^3*e^3*g^4*x+2816*a^3*c^2*d^2*e^4*f*g^3*x-3168*a^2*c^3*d^4*e^2*f*g^3*x-475
2*a^2*c^3*d^3*e^3*f^2*g^2*x+5544*a*c^4*d^5*e*f^2*g^2*x+3696*a*c^4*d^4*e^2*f^3*g*x-4620*c^5*d^6*f^3*g*x-1155*c^
5*d^5*e*f^4*x+1280*a^5*e^6*g^4-1408*a^4*c*d^2*e^4*g^4-5632*a^4*c*d*e^5*f*g^3+6336*a^3*c^2*d^3*e^3*f*g^3+9504*a
^3*c^2*d^2*e^4*f^2*g^2-11088*a^2*c^3*d^4*e^2*f^2*g^2-7392*a^2*c^3*d^3*e^3*f^3*g+9240*a*c^4*d^5*e*f^3*g+2310*a*
c^4*d^4*e^2*f^4-3465*c^5*d^6*f^4)*(e*x+d)^(1/2)/c^6/d^6/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

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maxima [A]  time = 0.74, size = 693, normalized size = 1.38 \[ \frac {2 \, {\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{4}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {8 \, {\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{3} g}{15 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {4 \, {\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \, {\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} - {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f^{2} g^{2}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} + \frac {8 \, {\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \, {\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} - {\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \, {\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \, {\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} f g^{3}}{315 \, \sqrt {c d x + a e} c^{5} d^{5}} + \frac {2 \, {\left (315 \, c^{6} d^{6} e x^{6} + 1408 \, a^{5} c d^{2} e^{5} - 1280 \, a^{6} e^{7} + 35 \, {\left (11 \, c^{6} d^{7} - a c^{5} d^{5} e^{2}\right )} x^{5} - 5 \, {\left (11 \, a c^{5} d^{6} e - 10 \, a^{2} c^{4} d^{4} e^{3}\right )} x^{4} + 8 \, {\left (11 \, a^{2} c^{4} d^{5} e^{2} - 10 \, a^{3} c^{3} d^{3} e^{4}\right )} x^{3} - 16 \, {\left (11 \, a^{3} c^{3} d^{4} e^{3} - 10 \, a^{4} c^{2} d^{2} e^{5}\right )} x^{2} + 64 \, {\left (11 \, a^{4} c^{2} d^{3} e^{4} - 10 \, a^{5} c d e^{6}\right )} x\right )} g^{4}}{3465 \, \sqrt {c d x + a e} c^{6} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/3*(c^2*d^2*e*x^2 + 3*a*c*d^2*e - 2*a^2*e^3 + (3*c^2*d^3 - a*c*d*e^2)*x)*f^4/(sqrt(c*d*x + a*e)*c^2*d^2) + 8/
15*(3*c^3*d^3*e*x^3 - 10*a^2*c*d^2*e^2 + 8*a^3*e^4 + (5*c^3*d^4 - a*c^2*d^2*e^2)*x^2 - (5*a*c^2*d^3*e - 4*a^2*
c*d*e^3)*x)*f^3*g/(sqrt(c*d*x + a*e)*c^3*d^3) + 4/35*(15*c^4*d^4*e*x^4 + 56*a^3*c*d^2*e^3 - 48*a^4*e^5 + 3*(7*
c^4*d^5 - a*c^3*d^3*e^2)*x^3 - (7*a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*x^2 + 4*(7*a^2*c^2*d^3*e^2 - 6*a^3*c*d*e^4)
*x)*f^2*g^2/(sqrt(c*d*x + a*e)*c^4*d^4) + 8/315*(35*c^5*d^5*e*x^5 - 144*a^4*c*d^2*e^4 + 128*a^5*e^6 + 5*(9*c^5
*d^6 - a*c^4*d^4*e^2)*x^4 - (9*a*c^4*d^5*e - 8*a^2*c^3*d^3*e^3)*x^3 + 2*(9*a^2*c^3*d^4*e^2 - 8*a^3*c^2*d^2*e^4
)*x^2 - 8*(9*a^3*c^2*d^3*e^3 - 8*a^4*c*d*e^5)*x)*f*g^3/(sqrt(c*d*x + a*e)*c^5*d^5) + 2/3465*(315*c^6*d^6*e*x^6
+ 1408*a^5*c*d^2*e^5 - 1280*a^6*e^7 + 35*(11*c^6*d^7 - a*c^5*d^5*e^2)*x^5 - 5*(11*a*c^5*d^6*e - 10*a^2*c^4*d^
4*e^3)*x^4 + 8*(11*a^2*c^4*d^5*e^2 - 10*a^3*c^3*d^3*e^4)*x^3 - 16*(11*a^3*c^3*d^4*e^3 - 10*a^4*c^2*d^2*e^5)*x^
2 + 64*(11*a^4*c^2*d^3*e^4 - 10*a^5*c*d*e^6)*x)*g^4/(sqrt(c*d*x + a*e)*c^6*d^6)

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mupad [B]  time = 4.09, size = 653, normalized size = 1.30 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^4\,x^5\,\sqrt {d+e\,x}}{11\,c\,d}-\frac {\sqrt {d+e\,x}\,\left (2560\,a^5\,e^6\,g^4-2816\,a^4\,c\,d^2\,e^4\,g^4-11264\,a^4\,c\,d\,e^5\,f\,g^3+12672\,a^3\,c^2\,d^3\,e^3\,f\,g^3+19008\,a^3\,c^2\,d^2\,e^4\,f^2\,g^2-22176\,a^2\,c^3\,d^4\,e^2\,f^2\,g^2-14784\,a^2\,c^3\,d^3\,e^3\,f^3\,g+18480\,a\,c^4\,d^5\,e\,f^3\,g+4620\,a\,c^4\,d^4\,e^2\,f^4-6930\,c^5\,d^6\,f^4\right )}{3465\,c^6\,d^6\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (1280\,a^4\,c\,d\,e^5\,g^4-1408\,a^3\,c^2\,d^3\,e^3\,g^4-5632\,a^3\,c^2\,d^2\,e^4\,f\,g^3+6336\,a^2\,c^3\,d^4\,e^2\,f\,g^3+9504\,a^2\,c^3\,d^3\,e^3\,f^2\,g^2-11088\,a\,c^4\,d^5\,e\,f^2\,g^2-7392\,a\,c^4\,d^4\,e^2\,f^3\,g+9240\,c^5\,d^6\,f^3\,g+2310\,c^5\,d^5\,e\,f^4\right )}{3465\,c^6\,d^6\,e}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-960\,a^3\,c^2\,d^2\,e^4\,g^4+1056\,a^2\,c^3\,d^4\,e^2\,g^4+4224\,a^2\,c^3\,d^3\,e^3\,f\,g^3-4752\,a\,c^4\,d^5\,e\,f\,g^3-7128\,a\,c^4\,d^4\,e^2\,f^2\,g^2+8316\,c^5\,d^6\,f^2\,g^2+5544\,c^5\,d^5\,e\,f^3\,g\right )}{3465\,c^6\,d^6\,e}+\frac {4\,g^2\,x^3\,\sqrt {d+e\,x}\,\left (40\,a^2\,e^3\,g^2-44\,a\,c\,d^2\,e\,g^2-176\,a\,c\,d\,e^2\,f\,g+198\,c^2\,d^3\,f\,g+297\,c^2\,d^2\,e\,f^2\right )}{693\,c^3\,d^3\,e}+\frac {2\,g^3\,x^4\,\sqrt {d+e\,x}\,\left (11\,c\,g\,d^2+44\,c\,f\,d\,e-10\,a\,g\,e^2\right )}{99\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^4*(d + e*x)^(3/2))/(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)*((2*g^4*x^5*(d + e*x)^(1/2))/(11*c*d) - ((d + e*x)^(1/2)*(2560*
a^5*e^6*g^4 - 6930*c^5*d^6*f^4 + 4620*a*c^4*d^4*e^2*f^4 - 2816*a^4*c*d^2*e^4*g^4 - 14784*a^2*c^3*d^3*e^3*f^3*g
+ 12672*a^3*c^2*d^3*e^3*f*g^3 + 18480*a*c^4*d^5*e*f^3*g - 11264*a^4*c*d*e^5*f*g^3 - 22176*a^2*c^3*d^4*e^2*f^2
*g^2 + 19008*a^3*c^2*d^2*e^4*f^2*g^2))/(3465*c^6*d^6*e) + (x*(d + e*x)^(1/2)*(2310*c^5*d^5*e*f^4 + 9240*c^5*d^
6*f^3*g - 1408*a^3*c^2*d^3*e^3*g^4 + 1280*a^4*c*d*e^5*g^4 - 7392*a*c^4*d^4*e^2*f^3*g - 11088*a*c^4*d^5*e*f^2*g
^2 + 6336*a^2*c^3*d^4*e^2*f*g^3 - 5632*a^3*c^2*d^2*e^4*f*g^3 + 9504*a^2*c^3*d^3*e^3*f^2*g^2))/(3465*c^6*d^6*e)
+ (x^2*(d + e*x)^(1/2)*(8316*c^5*d^6*f^2*g^2 + 1056*a^2*c^3*d^4*e^2*g^4 - 960*a^3*c^2*d^2*e^4*g^4 + 5544*c^5*
d^5*e*f^3*g - 7128*a*c^4*d^4*e^2*f^2*g^2 + 4224*a^2*c^3*d^3*e^3*f*g^3 - 4752*a*c^4*d^5*e*f*g^3))/(3465*c^6*d^6
*e) + (4*g^2*x^3*(d + e*x)^(1/2)*(40*a^2*e^3*g^2 + 297*c^2*d^2*e*f^2 + 198*c^2*d^3*f*g - 44*a*c*d^2*e*g^2 - 17
6*a*c*d*e^2*f*g))/(693*c^3*d^3*e) + (2*g^3*x^4*(d + e*x)^(1/2)*(11*c*d^2*g - 10*a*e^2*g + 44*c*d*e*f))/(99*c^2
*d^2*e)))/(x + d/e)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(g*x+f)**4/(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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