3.849 \(\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} (d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=189 \[ \frac{32 b \sqrt{a+b x} \left (35 a^2 e^2-84 a b d e+58 b^2 d^2\right )}{105 \sqrt{d+e x} (b d-a e)^4}+\frac{16 \sqrt{a+b x} \left (35 a^2 e^2-84 a b d e+58 b^2 d^2\right )}{105 (d+e x)^{3/2} (b d-a e)^3}+\frac{6 d^2 \sqrt{a+b x}}{7 (d+e x)^{7/2} (b d-a e)}+\frac{4 d \sqrt{a+b x} (23 b d-14 a e)}{35 (d+e x)^{5/2} (b d-a e)^2} \]

[Out]

(6*d^2*Sqrt[a + b*x])/(7*(b*d - a*e)*(d + e*x)^(7/2)) + (4*d*(23*b*d - 14*a*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^
2*(d + e*x)^(5/2)) + (16*(58*b^2*d^2 - 84*a*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x])/(105*(b*d - a*e)^3*(d + e*x)^(3
/2)) + (32*b*(58*b^2*d^2 - 84*a*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x])/(105*(b*d - a*e)^4*Sqrt[d + e*x])

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Rubi [A]  time = 0.19222, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {949, 78, 45, 37} \[ \frac{32 b \sqrt{a+b x} \left (35 a^2 e^2-84 a b d e+58 b^2 d^2\right )}{105 \sqrt{d+e x} (b d-a e)^4}+\frac{16 \sqrt{a+b x} \left (35 a^2 e^2-84 a b d e+58 b^2 d^2\right )}{105 (d+e x)^{3/2} (b d-a e)^3}+\frac{6 d^2 \sqrt{a+b x}}{7 (d+e x)^{7/2} (b d-a e)}+\frac{4 d \sqrt{a+b x} (23 b d-14 a e)}{35 (d+e x)^{5/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Int[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]

[Out]

(6*d^2*Sqrt[a + b*x])/(7*(b*d - a*e)*(d + e*x)^(7/2)) + (4*d*(23*b*d - 14*a*e)*Sqrt[a + b*x])/(35*(b*d - a*e)^
2*(d + e*x)^(5/2)) + (16*(58*b^2*d^2 - 84*a*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x])/(105*(b*d - a*e)^3*(d + e*x)^(3
/2)) + (32*b*(58*b^2*d^2 - 84*a*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x])/(105*(b*d - a*e)^4*Sqrt[d + e*x])

Rule 949

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[(R*(d + e*x)^(m + 1)*(f + g*x)^(n + 1))/((m + 1)*(e*f - d*g)), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} (d+e x)^{9/2}} \, dx &=\frac{6 d^2 \sqrt{a+b x}}{7 (b d-a e) (d+e x)^{7/2}}+\frac{2 \int \frac{3 d (17 b d-14 a e)+28 e (b d-a e) x}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx}{7 (b d-a e)}\\ &=\frac{6 d^2 \sqrt{a+b x}}{7 (b d-a e) (d+e x)^{7/2}}+\frac{4 d (23 b d-14 a e) \sqrt{a+b x}}{35 (b d-a e)^2 (d+e x)^{5/2}}+\frac{\left (8 \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right )\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx}{35 (b d-a e)^2}\\ &=\frac{6 d^2 \sqrt{a+b x}}{7 (b d-a e) (d+e x)^{7/2}}+\frac{4 d (23 b d-14 a e) \sqrt{a+b x}}{35 (b d-a e)^2 (d+e x)^{5/2}}+\frac{16 \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right ) \sqrt{a+b x}}{105 (b d-a e)^3 (d+e x)^{3/2}}+\frac{\left (16 b \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right )\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{105 (b d-a e)^3}\\ &=\frac{6 d^2 \sqrt{a+b x}}{7 (b d-a e) (d+e x)^{7/2}}+\frac{4 d (23 b d-14 a e) \sqrt{a+b x}}{35 (b d-a e)^2 (d+e x)^{5/2}}+\frac{16 \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right ) \sqrt{a+b x}}{105 (b d-a e)^3 (d+e x)^{3/2}}+\frac{32 b \left (58 b^2 d^2-84 a b d e+35 a^2 e^2\right ) \sqrt{a+b x}}{105 (b d-a e)^4 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.124592, size = 173, normalized size = 0.92 \[ \frac{2 \sqrt{a+b x} \left (a^2 b e^2 \left (3890 d^2 e x+1953 d^3+2632 d e^2 x^2+560 e^3 x^3\right )-a^3 e^3 \left (409 d^2+644 d e x+280 e^2 x^2\right )-a b^2 d e \left (6664 d^2 e x+2975 d^3+5168 d e^2 x^2+1344 e^3 x^3\right )+b^3 d^2 \left (3850 d^2 e x+1575 d^3+3248 d e^2 x^2+928 e^3 x^3\right )\right )}{105 (d+e x)^{7/2} (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]

[Out]

(2*Sqrt[a + b*x]*(-(a^3*e^3*(409*d^2 + 644*d*e*x + 280*e^2*x^2)) + a^2*b*e^2*(1953*d^3 + 3890*d^2*e*x + 2632*d
*e^2*x^2 + 560*e^3*x^3) + b^3*d^2*(1575*d^3 + 3850*d^2*e*x + 3248*d*e^2*x^2 + 928*e^3*x^3) - a*b^2*d*e*(2975*d
^3 + 6664*d^2*e*x + 5168*d*e^2*x^2 + 1344*e^3*x^3)))/(105*(b*d - a*e)^4*(d + e*x)^(7/2))

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Maple [A]  time = 0.057, size = 248, normalized size = 1.3 \begin{align*} -{\frac{-1120\,{a}^{2}b{e}^{5}{x}^{3}+2688\,a{b}^{2}d{e}^{4}{x}^{3}-1856\,{b}^{3}{d}^{2}{e}^{3}{x}^{3}+560\,{a}^{3}{e}^{5}{x}^{2}-5264\,{a}^{2}bd{e}^{4}{x}^{2}+10336\,a{b}^{2}{d}^{2}{e}^{3}{x}^{2}-6496\,{b}^{3}{d}^{3}{e}^{2}{x}^{2}+1288\,{a}^{3}d{e}^{4}x-7780\,{a}^{2}b{d}^{2}{e}^{3}x+13328\,a{b}^{2}{d}^{3}{e}^{2}x-7700\,{b}^{3}{d}^{4}ex+818\,{a}^{3}{d}^{2}{e}^{3}-3906\,{a}^{2}b{d}^{3}{e}^{2}+5950\,a{b}^{2}{d}^{4}e-3150\,{b}^{3}{d}^{5}}{105\,{e}^{4}{a}^{4}-420\,b{e}^{3}d{a}^{3}+630\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-420\,a{b}^{3}{d}^{3}e+105\,{b}^{4}{d}^{4}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x)

[Out]

-2/105*(b*x+a)^(1/2)*(-560*a^2*b*e^5*x^3+1344*a*b^2*d*e^4*x^3-928*b^3*d^2*e^3*x^3+280*a^3*e^5*x^2-2632*a^2*b*d
*e^4*x^2+5168*a*b^2*d^2*e^3*x^2-3248*b^3*d^3*e^2*x^2+644*a^3*d*e^4*x-3890*a^2*b*d^2*e^3*x+6664*a*b^2*d^3*e^2*x
-3850*b^3*d^4*e*x+409*a^3*d^2*e^3-1953*a^2*b*d^3*e^2+2975*a*b^2*d^4*e-1575*b^3*d^5)/(e*x+d)^(7/2)/(a^4*e^4-4*a
^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 14.2424, size = 1021, normalized size = 5.4 \begin{align*} \frac{2 \,{\left (1575 \, b^{3} d^{5} - 2975 \, a b^{2} d^{4} e + 1953 \, a^{2} b d^{3} e^{2} - 409 \, a^{3} d^{2} e^{3} + 16 \,{\left (58 \, b^{3} d^{2} e^{3} - 84 \, a b^{2} d e^{4} + 35 \, a^{2} b e^{5}\right )} x^{3} + 8 \,{\left (406 \, b^{3} d^{3} e^{2} - 646 \, a b^{2} d^{2} e^{3} + 329 \, a^{2} b d e^{4} - 35 \, a^{3} e^{5}\right )} x^{2} + 2 \,{\left (1925 \, b^{3} d^{4} e - 3332 \, a b^{2} d^{3} e^{2} + 1945 \, a^{2} b d^{2} e^{3} - 322 \, a^{3} d e^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{105 \,{\left (b^{4} d^{8} - 4 \, a b^{3} d^{7} e + 6 \, a^{2} b^{2} d^{6} e^{2} - 4 \, a^{3} b d^{5} e^{3} + a^{4} d^{4} e^{4} +{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{4} + 4 \,{\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{3} + 6 \,{\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x^{2} + 4 \,{\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/105*(1575*b^3*d^5 - 2975*a*b^2*d^4*e + 1953*a^2*b*d^3*e^2 - 409*a^3*d^2*e^3 + 16*(58*b^3*d^2*e^3 - 84*a*b^2*
d*e^4 + 35*a^2*b*e^5)*x^3 + 8*(406*b^3*d^3*e^2 - 646*a*b^2*d^2*e^3 + 329*a^2*b*d*e^4 - 35*a^3*e^5)*x^2 + 2*(19
25*b^3*d^4*e - 3332*a*b^2*d^3*e^2 + 1945*a^2*b*d^2*e^3 - 322*a^3*d*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^4*d^
8 - 4*a*b^3*d^7*e + 6*a^2*b^2*d^6*e^2 - 4*a^3*b*d^5*e^3 + a^4*d^4*e^4 + (b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 6*a^2
*b^2*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*x^4 + 4*(b^4*d^5*e^3 - 4*a*b^3*d^4*e^4 + 6*a^2*b^2*d^3*e^5 - 4*a^3*b*d
^2*e^6 + a^4*d*e^7)*x^3 + 6*(b^4*d^6*e^2 - 4*a*b^3*d^5*e^3 + 6*a^2*b^2*d^4*e^4 - 4*a^3*b*d^3*e^5 + a^4*d^2*e^6
)*x^2 + 4*(b^4*d^7*e - 4*a*b^3*d^6*e^2 + 6*a^2*b^2*d^5*e^3 - 4*a^3*b*d^4*e^4 + a^4*d^3*e^5)*x)

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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((8*e**2*x**2+20*d*e*x+15*d**2)/(e*x+d)**(9/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.45319, size = 707, normalized size = 3.74 \begin{align*} \frac{2 \,{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (58 \, b^{10} d^{2} e^{6} - 84 \, a b^{9} d e^{7} + 35 \, a^{2} b^{8} e^{8}\right )}{\left (b x + a\right )}}{b^{6} d^{4}{\left | b \right |} e^{3} - 4 \, a b^{5} d^{3}{\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2}{\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d{\left | b \right |} e^{6} + a^{4} b^{2}{\left | b \right |} e^{7}} + \frac{7 \,{\left (58 \, b^{11} d^{3} e^{5} - 142 \, a b^{10} d^{2} e^{6} + 119 \, a^{2} b^{9} d e^{7} - 35 \, a^{3} b^{8} e^{8}\right )}}{b^{6} d^{4}{\left | b \right |} e^{3} - 4 \, a b^{5} d^{3}{\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2}{\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d{\left | b \right |} e^{6} + a^{4} b^{2}{\left | b \right |} e^{7}}\right )} + \frac{35 \,{\left (55 \, b^{12} d^{4} e^{4} - 188 \, a b^{11} d^{3} e^{5} + 243 \, a^{2} b^{10} d^{2} e^{6} - 142 \, a^{3} b^{9} d e^{7} + 32 \, a^{4} b^{8} e^{8}\right )}}{b^{6} d^{4}{\left | b \right |} e^{3} - 4 \, a b^{5} d^{3}{\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2}{\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d{\left | b \right |} e^{6} + a^{4} b^{2}{\left | b \right |} e^{7}}\right )}{\left (b x + a\right )} + \frac{105 \,{\left (15 \, b^{13} d^{5} e^{3} - 65 \, a b^{12} d^{4} e^{4} + 113 \, a^{2} b^{11} d^{3} e^{5} - 99 \, a^{3} b^{10} d^{2} e^{6} + 44 \, a^{4} b^{9} d e^{7} - 8 \, a^{5} b^{8} e^{8}\right )}}{b^{6} d^{4}{\left | b \right |} e^{3} - 4 \, a b^{5} d^{3}{\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2}{\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d{\left | b \right |} e^{6} + a^{4} b^{2}{\left | b \right |} e^{7}}\right )} \sqrt{b x + a}}{105 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/105*(2*(4*(b*x + a)*(2*(58*b^10*d^2*e^6 - 84*a*b^9*d*e^7 + 35*a^2*b^8*e^8)*(b*x + a)/(b^6*d^4*abs(b)*e^3 - 4
*a*b^5*d^3*abs(b)*e^4 + 6*a^2*b^4*d^2*abs(b)*e^5 - 4*a^3*b^3*d*abs(b)*e^6 + a^4*b^2*abs(b)*e^7) + 7*(58*b^11*d
^3*e^5 - 142*a*b^10*d^2*e^6 + 119*a^2*b^9*d*e^7 - 35*a^3*b^8*e^8)/(b^6*d^4*abs(b)*e^3 - 4*a*b^5*d^3*abs(b)*e^4
+ 6*a^2*b^4*d^2*abs(b)*e^5 - 4*a^3*b^3*d*abs(b)*e^6 + a^4*b^2*abs(b)*e^7)) + 35*(55*b^12*d^4*e^4 - 188*a*b^11
*d^3*e^5 + 243*a^2*b^10*d^2*e^6 - 142*a^3*b^9*d*e^7 + 32*a^4*b^8*e^8)/(b^6*d^4*abs(b)*e^3 - 4*a*b^5*d^3*abs(b)
*e^4 + 6*a^2*b^4*d^2*abs(b)*e^5 - 4*a^3*b^3*d*abs(b)*e^6 + a^4*b^2*abs(b)*e^7))*(b*x + a) + 105*(15*b^13*d^5*e
^3 - 65*a*b^12*d^4*e^4 + 113*a^2*b^11*d^3*e^5 - 99*a^3*b^10*d^2*e^6 + 44*a^4*b^9*d*e^7 - 8*a^5*b^8*e^8)/(b^6*d
^4*abs(b)*e^3 - 4*a*b^5*d^3*abs(b)*e^4 + 6*a^2*b^4*d^2*abs(b)*e^5 - 4*a^3*b^3*d*abs(b)*e^6 + a^4*b^2*abs(b)*e^
7))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(7/2)