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

[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])

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Rubi [A]  time = 0.893591, antiderivative size = 501, normalized size of antiderivative = 1., 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 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]

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 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 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]

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*}

Mathematica [A]  time = 0.436734, 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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Maple [A]  time = 0.055, size = 641, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \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}ef{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}ef{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}cd{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}gx-4620\,{c}^{5}{d}^{6}{f}^{3}gx-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}cd{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 ) }{3465\,{c}^{6}{d}^{6}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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]

-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 = 1.71883, size = 936, normalized size = 1.87 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.46703, size = 1246, normalized size = 2.49 \begin{align*} \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 )}} \end{align*}

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)