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

Optimal. Leaf size=321 \[ -\frac{2 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g+c d (e f-7 d g)\right )}{35 c^2 d^2 g \sqrt{d+e x}}-\frac{8 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )}{105 c^3 d^3 e}+\frac{8 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{105 c^4 d^4 e g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d g \sqrt{d+e x}} \]

[Out]

(8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*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])/(105*c^4*d^4*e*g*Sqrt[d + e*x]) - (8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*d*g))*Sqrt[d
+ e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(105*c^3*d^3*e) - (2*(6*a*e^2*g + c*d*(e*f - 7*d*g))*(f +
g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^2*d^2*g*Sqrt[d + e*x]) + (2*e*(f + g*x)^3*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*c*d*g*Sqrt[d + e*x])

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Rubi [A]  time = 0.419845, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {880, 870, 794, 648} \[ -\frac{2 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g+c d (e f-7 d g)\right )}{35 c^2 d^2 g \sqrt{d+e x}}-\frac{8 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )}{105 c^3 d^3 e}+\frac{8 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{105 c^4 d^4 e g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d g \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*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])/(105*c^4*d^4*e*g*Sqrt[d + e*x]) - (8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*d*g))*Sqrt[d
+ e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(105*c^3*d^3*e) - (2*(6*a*e^2*g + c*d*(e*f - 7*d*g))*(f +
g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^2*d^2*g*Sqrt[d + e*x]) + (2*e*(f + g*x)^3*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*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)^2}{\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)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt{d+e x}}-\frac{1}{7} \left (-7 d+\frac{6 a e^2}{c d}+\frac{e f}{g}\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\\ &=-\frac{2 \left (6 a e^2 g+c d (e f-7 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}}{35 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt{d+e x}}-\frac{\left (4 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}{35 c^2 d^2 g}\\ &=-\frac{8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}}{105 c^3 d^3 e}-\frac{2 \left (6 a e^2 g+c d (e f-7 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}}{35 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt{d+e x}}+\frac{\left (4 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}{105 c^3 d^3 e g}\\ &=\frac{8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}}{105 c^4 d^4 e g \sqrt{d+e x}}-\frac{8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 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}}{105 c^3 d^3 e}-\frac{2 \left (6 a e^2 g+c d (e f-7 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}}{35 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.18984, size = 169, normalized size = 0.53 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 c d e^2 g (7 d g+14 e f+3 e g x)-48 a^3 e^4 g^2-2 a c^2 d^2 e \left (14 d g (5 f+g x)+e \left (35 f^2+28 f g x+9 g^2 x^2\right )\right )+c^3 d^3 \left (7 d \left (15 f^2+10 f g x+3 g^2 x^2\right )+e x \left (35 f^2+42 f g x+15 g^2 x^2\right )\right )\right )}{105 c^4 d^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(3/2)*(f + g*x)^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)]*(-48*a^3*e^4*g^2 + 8*a^2*c*d*e^2*g*(14*e*f + 7*d*g + 3*e*g*x) - 2*a*c^2*d^2*e
*(14*d*g*(5*f + g*x) + e*(35*f^2 + 28*f*g*x + 9*g^2*x^2)) + c^3*d^3*(7*d*(15*f^2 + 10*f*g*x + 3*g^2*x^2) + e*x
*(35*f^2 + 42*f*g*x + 15*g^2*x^2))))/(105*c^4*d^4*Sqrt[d + e*x])

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Maple [A]  time = 0.053, size = 255, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -15\,e{g}^{2}{x}^{3}{c}^{3}{d}^{3}+18\,a{c}^{2}{d}^{2}{e}^{2}{g}^{2}{x}^{2}-21\,{c}^{3}{d}^{4}{g}^{2}{x}^{2}-42\,{c}^{3}{d}^{3}efg{x}^{2}-24\,{a}^{2}cd{e}^{3}{g}^{2}x+28\,a{c}^{2}{d}^{3}e{g}^{2}x+56\,a{c}^{2}{d}^{2}{e}^{2}fgx-70\,{c}^{3}{d}^{4}fgx-35\,{c}^{3}{d}^{3}e{f}^{2}x+48\,{a}^{3}{e}^{4}{g}^{2}-56\,{a}^{2}c{d}^{2}{e}^{2}{g}^{2}-112\,{a}^{2}cd{e}^{3}fg+140\,a{c}^{2}{d}^{3}efg+70\,a{c}^{2}{d}^{2}{e}^{2}{f}^{2}-105\,{d}^{4}{f}^{2}{c}^{3} \right ) }{105\,{c}^{4}{d}^{4}}\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)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-2/105*(c*d*x+a*e)*(-15*c^3*d^3*e*g^2*x^3+18*a*c^2*d^2*e^2*g^2*x^2-21*c^3*d^4*g^2*x^2-42*c^3*d^3*e*f*g*x^2-24*
a^2*c*d*e^3*g^2*x+28*a*c^2*d^3*e*g^2*x+56*a*c^2*d^2*e^2*f*g*x-70*c^3*d^4*f*g*x-35*c^3*d^3*e*f^2*x+48*a^3*e^4*g
^2-56*a^2*c*d^2*e^2*g^2-112*a^2*c*d*e^3*f*g+140*a*c^2*d^3*e*f*g+70*a*c^2*d^2*e^2*f^2-105*c^3*d^4*f^2)*(e*x+d)^
(1/2)/c^4/d^4/(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 = 1.45183, size = 417, normalized size = 1.3 \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^{2}}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} + \frac{4 \,{\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 g}{15 \, \sqrt{c d x + a e} c^{3} d^{3}} + \frac{2 \,{\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 )} g^{2}}{105 \, \sqrt{c d x + a e} c^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^2/(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^2/(sqrt(c*d*x + a*e)*c^2*d^2) + 4/
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*g/(sqrt(c*d*x + a*e)*c^3*d^3) + 2/105*(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)*g^2/(sqrt(c*d*x + a*e)*c^4*d^4)

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Fricas [A]  time = 1.65022, size = 533, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (15 \, c^{3} d^{3} e g^{2} x^{3} + 35 \,{\left (3 \, c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2}\right )} f^{2} - 28 \,{\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} f g + 8 \,{\left (7 \, a^{2} c d^{2} e^{2} - 6 \, a^{3} e^{4}\right )} g^{2} + 3 \,{\left (14 \, c^{3} d^{3} e f g +{\left (7 \, c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2}\right )} g^{2}\right )} x^{2} +{\left (35 \, c^{3} d^{3} e f^{2} + 14 \,{\left (5 \, c^{3} d^{4} - 4 \, a c^{2} d^{2} e^{2}\right )} f g - 4 \,{\left (7 \, a c^{2} d^{3} e - 6 \, a^{2} c d e^{3}\right )} g^{2}\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}}{105 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*c^3*d^3*e*g^2*x^3 + 35*(3*c^3*d^4 - 2*a*c^2*d^2*e^2)*f^2 - 28*(5*a*c^2*d^3*e - 4*a^2*c*d*e^3)*f*g +
8*(7*a^2*c*d^2*e^2 - 6*a^3*e^4)*g^2 + 3*(14*c^3*d^3*e*f*g + (7*c^3*d^4 - 6*a*c^2*d^2*e^2)*g^2)*x^2 + (35*c^3*d
^3*e*f^2 + 14*(5*c^3*d^4 - 4*a*c^2*d^2*e^2)*f*g - 4*(7*a*c^2*d^3*e - 6*a^2*c*d*e^3)*g^2)*x)*sqrt(c*d*e*x^2 + a
*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^4*d^4*e*x + c^4*d^5)

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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)**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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{2}}{\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)^2/(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)^2/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x), x)