3.848 \(\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=133 \[ \frac{16 \sqrt{a+b x} \left (15 a^2 e^2-35 a b d e+23 b^2 d^2\right )}{15 \sqrt{d+e x} (b d-a e)^3}+\frac{6 d^2 \sqrt{a+b x}}{5 (d+e x)^{5/2} (b d-a e)}+\frac{8 d \sqrt{a+b x} (8 b d-5 a e)}{15 (d+e x)^{3/2} (b d-a e)^2} \]

[Out]

(6*d^2*Sqrt[a + b*x])/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (8*d*(8*b*d - 5*a*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^2*
(d + e*x)^(3/2)) + (16*(23*b^2*d^2 - 35*a*b*d*e + 15*a^2*e^2)*Sqrt[a + b*x])/(15*(b*d - a*e)^3*Sqrt[d + e*x])

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Rubi [A]  time = 0.1296, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {949, 78, 37} \[ \frac{16 \sqrt{a+b x} \left (15 a^2 e^2-35 a b d e+23 b^2 d^2\right )}{15 \sqrt{d+e x} (b d-a e)^3}+\frac{6 d^2 \sqrt{a+b x}}{5 (d+e x)^{5/2} (b d-a e)}+\frac{8 d \sqrt{a+b x} (8 b d-5 a e)}{15 (d+e x)^{3/2} (b d-a e)^2} \]

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)^(7/2)),x]

[Out]

(6*d^2*Sqrt[a + b*x])/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (8*d*(8*b*d - 5*a*e)*Sqrt[a + b*x])/(15*(b*d - a*e)^2*
(d + e*x)^(3/2)) + (16*(23*b^2*d^2 - 35*a*b*d*e + 15*a^2*e^2)*Sqrt[a + b*x])/(15*(b*d - a*e)^3*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 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)^{7/2}} \, dx &=\frac{6 d^2 \sqrt{a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac{2 \int \frac{6 d (6 b d-5 a e)+20 e (b d-a e) x}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx}{5 (b d-a e)}\\ &=\frac{6 d^2 \sqrt{a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac{8 d (8 b d-5 a e) \sqrt{a+b x}}{15 (b d-a e)^2 (d+e x)^{3/2}}+\frac{\left (8 \left (23 b^2 d^2-35 a b d e+15 a^2 e^2\right )\right ) \int \frac{1}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx}{15 (b d-a e)^2}\\ &=\frac{6 d^2 \sqrt{a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac{8 d (8 b d-5 a e) \sqrt{a+b x}}{15 (b d-a e)^2 (d+e x)^{3/2}}+\frac{16 \left (23 b^2 d^2-35 a b d e+15 a^2 e^2\right ) \sqrt{a+b x}}{15 (b d-a e)^3 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0876009, size = 110, normalized size = 0.83 \[ \frac{2 \sqrt{a+b x} \left (a^2 e^2 \left (149 d^2+260 d e x+120 e^2 x^2\right )-2 a b d e \left (175 d^2+306 d e x+140 e^2 x^2\right )+b^2 d^2 \left (225 d^2+400 d e x+184 e^2 x^2\right )\right )}{15 (d+e x)^{5/2} (b d-a e)^3} \]

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)^(7/2)),x]

[Out]

(2*Sqrt[a + b*x]*(a^2*e^2*(149*d^2 + 260*d*e*x + 120*e^2*x^2) - 2*a*b*d*e*(175*d^2 + 306*d*e*x + 140*e^2*x^2)
+ b^2*d^2*(225*d^2 + 400*d*e*x + 184*e^2*x^2)))/(15*(b*d - a*e)^3*(d + e*x)^(5/2))

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Maple [A]  time = 0.055, size = 150, normalized size = 1.1 \begin{align*} -{\frac{240\,{a}^{2}{e}^{4}{x}^{2}-560\,abd{e}^{3}{x}^{2}+368\,{b}^{2}{d}^{2}{e}^{2}{x}^{2}+520\,{a}^{2}d{e}^{3}x-1224\,ab{d}^{2}{e}^{2}x+800\,{b}^{2}{d}^{3}ex+298\,{a}^{2}{d}^{2}{e}^{2}-700\,ab{d}^{3}e+450\,{b}^{2}{d}^{4}}{15\,{a}^{3}{e}^{3}-45\,{a}^{2}bd{e}^{2}+45\,a{b}^{2}{d}^{2}e-15\,{b}^{3}{d}^{3}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{5}{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)^(7/2)/(b*x+a)^(1/2),x)

[Out]

-2/15*(b*x+a)^(1/2)*(120*a^2*e^4*x^2-280*a*b*d*e^3*x^2+184*b^2*d^2*e^2*x^2+260*a^2*d*e^3*x-612*a*b*d^2*e^2*x+4
00*b^2*d^3*e*x+149*a^2*d^2*e^2-350*a*b*d^3*e+225*b^2*d^4)/(e*x+d)^(5/2)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b
^3*d^3)

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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)^(7/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.95919, size = 608, normalized size = 4.57 \begin{align*} \frac{2 \,{\left (225 \, b^{2} d^{4} - 350 \, a b d^{3} e + 149 \, a^{2} d^{2} e^{2} + 8 \,{\left (23 \, b^{2} d^{2} e^{2} - 35 \, a b d e^{3} + 15 \, a^{2} e^{4}\right )} x^{2} + 4 \,{\left (100 \, b^{2} d^{3} e - 153 \, a b d^{2} e^{2} + 65 \, a^{2} d e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} +{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \,{\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\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)^(7/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/15*(225*b^2*d^4 - 350*a*b*d^3*e + 149*a^2*d^2*e^2 + 8*(23*b^2*d^2*e^2 - 35*a*b*d*e^3 + 15*a^2*e^4)*x^2 + 4*(
100*b^2*d^3*e - 153*a*b*d^2*e^2 + 65*a^2*d*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^3*d^6 - 3*a*b^2*d^5*e + 3*a^
2*b*d^4*e^2 - a^3*d^3*e^3 + (b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6)*x^3 + 3*(b^3*d^4*e^2 - 3
*a*b^2*d^3*e^3 + 3*a^2*b*d^2*e^4 - a^3*d*e^5)*x^2 + 3*(b^3*d^5*e - 3*a*b^2*d^4*e^2 + 3*a^2*b*d^3*e^3 - a^3*d^2
*e^4)*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)**(7/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.29768, size = 455, normalized size = 3.42 \begin{align*} \frac{2 \,{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (23 \, b^{8} d^{2} e^{4} - 35 \, a b^{7} d e^{5} + 15 \, a^{2} b^{6} e^{6}\right )}{\left (b x + a\right )}}{b^{5} d^{3}{\left | b \right |} e^{2} - 3 \, a b^{4} d^{2}{\left | b \right |} e^{3} + 3 \, a^{2} b^{3} d{\left | b \right |} e^{4} - a^{3} b^{2}{\left | b \right |} e^{5}} + \frac{5 \,{\left (20 \, b^{9} d^{3} e^{3} - 49 \, a b^{8} d^{2} e^{4} + 41 \, a^{2} b^{7} d e^{5} - 12 \, a^{3} b^{6} e^{6}\right )}}{b^{5} d^{3}{\left | b \right |} e^{2} - 3 \, a b^{4} d^{2}{\left | b \right |} e^{3} + 3 \, a^{2} b^{3} d{\left | b \right |} e^{4} - a^{3} b^{2}{\left | b \right |} e^{5}}\right )} + \frac{15 \,{\left (15 \, b^{10} d^{4} e^{2} - 50 \, a b^{9} d^{3} e^{3} + 63 \, a^{2} b^{8} d^{2} e^{4} - 36 \, a^{3} b^{7} d e^{5} + 8 \, a^{4} b^{6} e^{6}\right )}}{b^{5} d^{3}{\left | b \right |} e^{2} - 3 \, a b^{4} d^{2}{\left | b \right |} e^{3} + 3 \, a^{2} b^{3} d{\left | b \right |} e^{4} - a^{3} b^{2}{\left | b \right |} e^{5}}\right )} \sqrt{b x + a}}{15 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{5}{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)^(7/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/15*(4*(b*x + a)*(2*(23*b^8*d^2*e^4 - 35*a*b^7*d*e^5 + 15*a^2*b^6*e^6)*(b*x + a)/(b^5*d^3*abs(b)*e^2 - 3*a*b^
4*d^2*abs(b)*e^3 + 3*a^2*b^3*d*abs(b)*e^4 - a^3*b^2*abs(b)*e^5) + 5*(20*b^9*d^3*e^3 - 49*a*b^8*d^2*e^4 + 41*a^
2*b^7*d*e^5 - 12*a^3*b^6*e^6)/(b^5*d^3*abs(b)*e^2 - 3*a*b^4*d^2*abs(b)*e^3 + 3*a^2*b^3*d*abs(b)*e^4 - a^3*b^2*
abs(b)*e^5)) + 15*(15*b^10*d^4*e^2 - 50*a*b^9*d^3*e^3 + 63*a^2*b^8*d^2*e^4 - 36*a^3*b^7*d*e^5 + 8*a^4*b^6*e^6)
/(b^5*d^3*abs(b)*e^2 - 3*a*b^4*d^2*abs(b)*e^3 + 3*a^2*b^3*d*abs(b)*e^4 - a^3*b^2*abs(b)*e^5))*sqrt(b*x + a)/(b
^2*d + (b*x + a)*b*e - a*b*e)^(5/2)