3.6.7 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)} \]

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Rubi [A]  time = 0.22, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {872, 860} \begin {gather*} \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(9/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(7/2)) + (8*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (16*c
^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(3/2))

Rule 860

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

Rule 872

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {(4 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx}{7 (c d f-a e g)}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {\left (8 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx}{35 (c d f-a e g)^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 105, normalized size = 0.53 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (15 a^2 e^2 g^2-6 a c d e g (7 f+2 g x)+c^2 d^2 \left (35 f^2+28 f g x+8 g^2 x^2\right )\right )}{105 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(9/2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 8.34, size = 137, normalized size = 0.69 \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {35 c^2 d^2 (a e+c d x)^{3/2}}{(f+g x)^{3/2}}+\frac {15 g^2 (a e+c d x)^{7/2}}{(f+g x)^{7/2}}-\frac {42 c d g (a e+c d x)^{5/2}}{(f+g x)^{5/2}}\right )}{105 \sqrt {(d+e x) (a e+c d x)} (c d f-a e g)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(9/2)),x]

[Out]

(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*((15*g^2*(a*e + c*d*x)^(7/2))/(f + g*x)^(7/2) - (42*c*d*g*(a*e + c*d*x)^(5/
2))/(f + g*x)^(5/2) + (35*c^2*d^2*(a*e + c*d*x)^(3/2))/(f + g*x)^(3/2)))/(105*(c*d*f - a*e*g)^3*Sqrt[(a*e + c*
d*x)*(d + e*x)])

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fricas [B]  time = 0.45, size = 748, normalized size = 3.78 \begin {gather*} \frac {2 \, {\left (8 \, c^{3} d^{3} g^{2} x^{3} + 35 \, a c^{2} d^{2} e f^{2} - 42 \, a^{2} c d e^{2} f g + 15 \, a^{3} e^{3} g^{2} + 4 \, {\left (7 \, c^{3} d^{3} f g - a c^{2} d^{2} e g^{2}\right )} x^{2} + {\left (35 \, c^{3} d^{3} f^{2} - 14 \, a c^{2} d^{2} e f g + 3 \, a^{2} c d e^{2} 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} \sqrt {g x + f}}{105 \, {\left (c^{3} d^{4} f^{7} - 3 \, a c^{2} d^{3} e f^{6} g + 3 \, a^{2} c d^{2} e^{2} f^{5} g^{2} - a^{3} d e^{3} f^{4} g^{3} + {\left (c^{3} d^{3} e f^{3} g^{4} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{5} + 3 \, a^{2} c d e^{3} f g^{6} - a^{3} e^{4} g^{7}\right )} x^{5} + {\left (4 \, c^{3} d^{3} e f^{4} g^{3} - a^{3} d e^{3} g^{7} + {\left (c^{3} d^{4} - 12 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{4} - 3 \, {\left (a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} f^{2} g^{5} + {\left (3 \, a^{2} c d^{2} e^{2} - 4 \, a^{3} e^{4}\right )} f g^{6}\right )} x^{4} + 2 \, {\left (3 \, c^{3} d^{3} e f^{5} g^{2} - 2 \, a^{3} d e^{3} f g^{6} + {\left (2 \, c^{3} d^{4} - 9 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{3} - 3 \, {\left (2 \, a c^{2} d^{3} e - 3 \, a^{2} c d e^{3}\right )} f^{3} g^{4} + 3 \, {\left (2 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{2} g^{5}\right )} x^{3} + 2 \, {\left (2 \, c^{3} d^{3} e f^{6} g - 3 \, a^{3} d e^{3} f^{2} g^{5} + 3 \, {\left (c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2}\right )} f^{5} g^{2} - 3 \, {\left (3 \, a c^{2} d^{3} e - 2 \, a^{2} c d e^{3}\right )} f^{4} g^{3} + {\left (9 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4}\right )} f^{3} g^{4}\right )} x^{2} + {\left (c^{3} d^{3} e f^{7} - 4 \, a^{3} d e^{3} f^{3} g^{4} + {\left (4 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{6} g - 3 \, {\left (4 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{5} g^{2} + {\left (12 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{4} g^{3}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/105*(8*c^3*d^3*g^2*x^3 + 35*a*c^2*d^2*e*f^2 - 42*a^2*c*d*e^2*f*g + 15*a^3*e^3*g^2 + 4*(7*c^3*d^3*f*g - a*c^2
*d^2*e*g^2)*x^2 + (35*c^3*d^3*f^2 - 14*a*c^2*d^2*e*f*g + 3*a^2*c*d*e^2*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)/(c^3*d^4*f^7 - 3*a*c^2*d^3*e*f^6*g + 3*a^2*c*d^2*e^2*f^5*g^2 - a^3*d*
e^3*f^4*g^3 + (c^3*d^3*e*f^3*g^4 - 3*a*c^2*d^2*e^2*f^2*g^5 + 3*a^2*c*d*e^3*f*g^6 - a^3*e^4*g^7)*x^5 + (4*c^3*d
^3*e*f^4*g^3 - a^3*d*e^3*g^7 + (c^3*d^4 - 12*a*c^2*d^2*e^2)*f^3*g^4 - 3*(a*c^2*d^3*e - 4*a^2*c*d*e^3)*f^2*g^5
+ (3*a^2*c*d^2*e^2 - 4*a^3*e^4)*f*g^6)*x^4 + 2*(3*c^3*d^3*e*f^5*g^2 - 2*a^3*d*e^3*f*g^6 + (2*c^3*d^4 - 9*a*c^2
*d^2*e^2)*f^4*g^3 - 3*(2*a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^3*g^4 + 3*(2*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^5)*x^3 + 2
*(2*c^3*d^3*e*f^6*g - 3*a^3*d*e^3*f^2*g^5 + 3*(c^3*d^4 - 2*a*c^2*d^2*e^2)*f^5*g^2 - 3*(3*a*c^2*d^3*e - 2*a^2*c
*d*e^3)*f^4*g^3 + (9*a^2*c*d^2*e^2 - 2*a^3*e^4)*f^3*g^4)*x^2 + (c^3*d^3*e*f^7 - 4*a^3*d*e^3*f^3*g^4 + (4*c^3*d
^4 - 3*a*c^2*d^2*e^2)*f^6*g - 3*(4*a*c^2*d^3*e - a^2*c*d*e^3)*f^5*g^2 + (12*a^2*c*d^2*e^2 - a^3*e^4)*f^4*g^3)*
x)

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giac [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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.01, size = 169, normalized size = 0.85 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-12 a c d e \,g^{2} x +28 c^{2} d^{2} f g x +15 a^{2} e^{2} g^{2}-42 a c d e f g +35 f^{2} c^{2} d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (g x +f \right )^{\frac {7}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \sqrt {e x +d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x)

[Out]

-2/105*(c*d*x+a*e)*(8*c^2*d^2*g^2*x^2-12*a*c*d*e*g^2*x+28*c^2*d^2*f*g*x+15*a^2*e^2*g^2-42*a*c*d*e*f*g+35*c^2*d
^2*f^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(g*x+f)^(7/2)/(a^3*e^3*g^3-3*a^2*c*d*e^2*f*g^2+3*a*c^2*d^2*e*f
^2*g-c^3*d^3*f^3)/(e*x+d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^(9/2)), x)

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mupad [B]  time = 4.29, size = 289, normalized size = 1.46 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {30\,a^3\,e^3\,g^2-84\,a^2\,c\,d\,e^2\,f\,g+70\,a\,c^2\,d^2\,e\,f^2}{105\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {x\,\left (6\,a^2\,c\,d\,e^2\,g^2-28\,a\,c^2\,d^2\,e\,f\,g+70\,c^3\,d^3\,f^2\right )}{105\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^3\,d^3\,x^3}{105\,g\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c^2\,d^2\,x^2\,\left (a\,e\,g-7\,c\,d\,f\right )}{105\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {3\,f\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {3\,f^2\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/((f + g*x)^(9/2)*(d + e*x)^(1/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((30*a^3*e^3*g^2 + 70*a*c^2*d^2*e*f^2 - 84*a^2*c*d*e^2*f*g)/(1
05*g^3*(a*e*g - c*d*f)^3) + (x*(70*c^3*d^3*f^2 + 6*a^2*c*d*e^2*g^2 - 28*a*c^2*d^2*e*f*g))/(105*g^3*(a*e*g - c*
d*f)^3) + (16*c^3*d^3*x^3)/(105*g*(a*e*g - c*d*f)^3) - (8*c^2*d^2*x^2*(a*e*g - 7*c*d*f))/(105*g^2*(a*e*g - c*d
*f)^3)))/(x^3*(f + g*x)^(1/2)*(d + e*x)^(1/2) + (f^3*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^3 + (3*f*x^2*(f + g*x)
^(1/2)*(d + e*x)^(1/2))/g + (3*f^2*x*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^2)

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**(9/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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