3.5.39 \(\int \frac {(d+e x)^{5/2} (f+g x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=239 \[ -\frac {16 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^3 d^3 e}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.28, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {866, 794, 648} \begin {gather*} -\frac {16 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^4 d^4 e \sqrt {d+e x}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^3 d^3 e}-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(5/2)*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^(3/2)*(f + g*x)^3)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (4*g*Sqrt[d + e*x]*(f
+ g*x)^2)/(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (16*g^2*(2*a*e^2*g - c*d*(3*e*f - d*g))*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^4*d^4*e*Sqrt[d + e*x]) + (16*g^3*Sqrt[d + e*x]*Sqrt[a*d*e + (c*
d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^3*d^3*e)

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 866

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*(p + 1)), x] - Dist[(e*g*n)/(c*(p + 1)), I
nt[(d + e*x)^(m - 1)*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, 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[p, -1] && GtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{5/2} (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(2 g) \int \frac {(d+e x)^{3/2} (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{c d}\\ &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (8 g^2\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}{c^2 d^2}\\ &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 e}-\frac {\left (8 g^2 \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}{3 c^3 d^3 e}\\ &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {16 g^2 \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}}{3 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 e}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 131, normalized size = 0.55 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (f-g x)-6 a c^2 d^2 e g \left (f^2-6 f g x+g^2 x^2\right )+c^3 d^3 \left (-f^3-9 f^2 g x+9 f g^2 x^2+g^3 x^3\right )\right )}{3 c^4 d^4 ((d+e x) (a e+c d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(5/2)*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(2*(d + e*x)^(3/2)*(-16*a^3*e^3*g^3 + 24*a^2*c*d*e^2*g^2*(f - g*x) - 6*a*c^2*d^2*e*g*(f^2 - 6*f*g*x + g^2*x^2)
+ c^3*d^3*(-f^3 - 9*f^2*g*x + 9*f*g^2*x^2 + g^3*x^3)))/(3*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(3/2))

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IntegrateAlgebraic [A]  time = 4.48, size = 201, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{5/2} (a e+c d x) \left (a^3 e^3 g^3-3 a^2 c d e^2 f g^2-9 a^2 e^2 g^3 (a e+c d x)-9 c^2 d^2 f^2 g (a e+c d x)+3 a c^2 d^2 e f^2 g+9 c d f g^2 (a e+c d x)^2+18 a c d e f g^2 (a e+c d x)+g^3 (a e+c d x)^3-9 a e g^3 (a e+c d x)^2-c^3 d^3 f^3\right )}{3 c^4 d^4 ((d+e x) (a e+c d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((d + e*x)^(5/2)*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(2*(a*e + c*d*x)*(d + e*x)^(5/2)*(-(c^3*d^3*f^3) + 3*a*c^2*d^2*e*f^2*g - 3*a^2*c*d*e^2*f*g^2 + a^3*e^3*g^3 - 9
*c^2*d^2*f^2*g*(a*e + c*d*x) + 18*a*c*d*e*f*g^2*(a*e + c*d*x) - 9*a^2*e^2*g^3*(a*e + c*d*x) + 9*c*d*f*g^2*(a*e
+ c*d*x)^2 - 9*a*e*g^3*(a*e + c*d*x)^2 + g^3*(a*e + c*d*x)^3))/(3*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(5/2))

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fricas [A]  time = 0.42, size = 251, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} g^{3} x^{3} - c^{3} d^{3} f^{3} - 6 \, a c^{2} d^{2} e f^{2} g + 24 \, a^{2} c d e^{2} f g^{2} - 16 \, a^{3} e^{3} g^{3} + 3 \, {\left (3 \, c^{3} d^{3} f g^{2} - 2 \, a c^{2} d^{2} e g^{3}\right )} x^{2} - 3 \, {\left (3 \, c^{3} d^{3} f^{2} g - 12 \, a c^{2} d^{2} e f g^{2} + 8 \, a^{2} c d e^{2} g^{3}\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}}{3 \, {\left (c^{6} d^{6} e x^{3} + a^{2} c^{4} d^{5} e^{2} + {\left (c^{6} d^{7} + 2 \, a c^{5} d^{5} e^{2}\right )} x^{2} + {\left (2 \, a c^{5} d^{6} e + a^{2} c^{4} d^{4} e^{3}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

2/3*(c^3*d^3*g^3*x^3 - c^3*d^3*f^3 - 6*a*c^2*d^2*e*f^2*g + 24*a^2*c*d*e^2*f*g^2 - 16*a^3*e^3*g^3 + 3*(3*c^3*d^
3*f*g^2 - 2*a*c^2*d^2*e*g^3)*x^2 - 3*(3*c^3*d^3*f^2*g - 12*a*c^2*d^2*e*f*g^2 + 8*a^2*c*d*e^2*g^3)*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^3 + a^2*c^4*d^5*e^2 + (c^6*d^7 + 2*a*c^5*d^5*e^2
)*x^2 + (2*a*c^5*d^6*e + a^2*c^4*d^4*e^3)*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 6.27Unable to transpose Error: Bad Argument Value

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maple [A]  time = 0.01, size = 187, normalized size = 0.78 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-g^{3} x^{3} c^{3} d^{3}+6 a \,c^{2} d^{2} e \,g^{3} x^{2}-9 c^{3} d^{3} f \,g^{2} x^{2}+24 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x +9 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-24 a^{2} c d \,e^{2} f \,g^{2}+6 a \,c^{2} d^{2} e \,f^{2} g +f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}} c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(g*x+f)^3/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2),x)

[Out]

-2/3*(c*d*x+a*e)*(-c^3*d^3*g^3*x^3+6*a*c^2*d^2*e*g^3*x^2-9*c^3*d^3*f*g^2*x^2+24*a^2*c*d*e^2*g^3*x-36*a*c^2*d^2
*e*f*g^2*x+9*c^3*d^3*f^2*g*x+16*a^3*e^3*g^3-24*a^2*c*d*e^2*f*g^2+6*a*c^2*d^2*e*f^2*g+c^3*d^3*f^3)*(e*x+d)^(5/2
)/c^4/d^4/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)

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maxima [A]  time = 0.74, size = 219, normalized size = 0.92 \begin {gather*} -\frac {2 \, {\left (3 \, c d x + 2 \, a e\right )} f^{2} g}{{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt {c d x + a e}} + \frac {2 \, {\left (3 \, c^{2} d^{2} x^{2} + 12 \, a c d e x + 8 \, a^{2} e^{2}\right )} f g^{2}}{{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )} \sqrt {c d x + a e}} + \frac {2 \, {\left (c^{3} d^{3} x^{3} - 6 \, a c^{2} d^{2} e x^{2} - 24 \, a^{2} c d e^{2} x - 16 \, a^{3} e^{3}\right )} g^{3}}{3 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )} \sqrt {c d x + a e}} - \frac {2 \, f^{3}}{3 \, {\left (c^{2} d^{2} x + a c d e\right )} \sqrt {c d x + a e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

-2*(3*c*d*x + 2*a*e)*f^2*g/((c^3*d^3*x + a*c^2*d^2*e)*sqrt(c*d*x + a*e)) + 2*(3*c^2*d^2*x^2 + 12*a*c*d*e*x + 8
*a^2*e^2)*f*g^2/((c^4*d^4*x + a*c^3*d^3*e)*sqrt(c*d*x + a*e)) + 2/3*(c^3*d^3*x^3 - 6*a*c^2*d^2*e*x^2 - 24*a^2*
c*d*e^2*x - 16*a^3*e^3)*g^3/((c^5*d^5*x + a*c^4*d^4*e)*sqrt(c*d*x + a*e)) - 2/3*f^3/((c^2*d^2*x + a*c*d*e)*sqr
t(c*d*x + a*e))

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mupad [B]  time = 3.77, size = 278, normalized size = 1.16 \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 (\frac {32\,a^3\,e^3\,g^3}{3}-16\,a^2\,c\,d\,e^2\,f\,g^2+4\,a\,c^2\,d^2\,e\,f^2\,g+\frac {2\,c^3\,d^3\,f^3}{3}\right )}{c^6\,d^6\,e}-\frac {2\,g^3\,x^3\,\sqrt {d+e\,x}}{3\,c^3\,d^3\,e}+\frac {g^2\,x^2\,\left (4\,a\,e\,g-6\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c^4\,d^4\,e}+\frac {2\,g\,x\,\sqrt {d+e\,x}\,\left (8\,a^2\,e^2\,g^2-12\,a\,c\,d\,e\,f\,g+3\,c^2\,d^2\,f^2\right )}{c^5\,d^5\,e}\right )}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (c^6\,d^7+2\,a\,c^5\,d^5\,e^2\right )}{c^6\,d^6\,e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^3*(d + e*x)^(5/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(((d + e*x)^(1/2)*((32*a^3*e^3*g^3)/3 + (2*c^3*d^3*f^3)/3 + 4*
a*c^2*d^2*e*f^2*g - 16*a^2*c*d*e^2*f*g^2))/(c^6*d^6*e) - (2*g^3*x^3*(d + e*x)^(1/2))/(3*c^3*d^3*e) + (g^2*x^2*
(4*a*e*g - 6*c*d*f)*(d + e*x)^(1/2))/(c^4*d^4*e) + (2*g*x*(d + e*x)^(1/2)*(8*a^2*e^2*g^2 + 3*c^2*d^2*f^2 - 12*
a*c*d*e*f*g))/(c^5*d^5*e)))/(x^3 + (a^2*e)/(c^2*d) + (a*x*(a*e^2 + 2*c*d^2))/(c^2*d^2) + (x^2*(c^6*d^7 + 2*a*c
^5*d^5*e^2))/(c^6*d^6*e))

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sympy [F(-1)]  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)**(5/2)*(g*x+f)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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