3.1747 \(\int \frac{(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx\)

Optimal. Leaf size=896 \[ \frac{8 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} d^3}{27 b^{5/3} (b c-a d)^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{8 \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d^2}{9\ 3^{3/4} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{4 \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d^2}{27 \sqrt [4]{3} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{8 (c+d x)^{5/6} d^2}{27 b (b c-a d)^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{5/6} d}{9 b (b c-a d) (a+b x)^{3/2}}-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}} \]

[Out]

(-2*(c + d*x)^(5/6))/(5*b*(a + b*x)^(5/2)) - (2*d*(c + d*x)^(5/6))/(9*b*(b*c - a*d)*(a + b*x)^(3/2)) + (8*d^2*
(c + d*x)^(5/6))/(27*b*(b*c - a*d)^2*Sqrt[a + b*x]) + (8*(1 + Sqrt[3])*d^3*Sqrt[a + b*x]*(c + d*x)^(1/6))/(27*
b^(5/3)*(b*c - a*d)^2*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) + (8*d^2*(c + d*x)^(1/6)*((
b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3
) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticE[ArcCos[(
(b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x
)^(1/3))], (2 + Sqrt[3])/4])/(9*3^(3/4)*b^(5/3)*(b*c - a*d)^(5/3)*Sqrt[a + b*x]*Sqrt[-((b^(1/3)*(c + d*x)^(1/3
)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2
)]) + (4*(1 - Sqrt[3])*d^2*(c + d*x)^(1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/
3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b
^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcCos[((b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c
- a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(27*3^(1/4)*b^(5/3)*(b*c - a*d)^(5/3
)*Sqrt[a + b*x]*Sqrt[-((b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1
/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)])

________________________________________________________________________________________

Rubi [A]  time = 0.90147, antiderivative size = 896, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {47, 51, 63, 308, 225, 1881} \[ \frac{8 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x} d^3}{27 b^{5/3} (b c-a d)^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{8 \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d^2}{9\ 3^{3/4} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{4 \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) d^2}{27 \sqrt [4]{3} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{8 (c+d x)^{5/6} d^2}{27 b (b c-a d)^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{5/6} d}{9 b (b c-a d) (a+b x)^{3/2}}-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/6)/(a + b*x)^(7/2),x]

[Out]

(-2*(c + d*x)^(5/6))/(5*b*(a + b*x)^(5/2)) - (2*d*(c + d*x)^(5/6))/(9*b*(b*c - a*d)*(a + b*x)^(3/2)) + (8*d^2*
(c + d*x)^(5/6))/(27*b*(b*c - a*d)^2*Sqrt[a + b*x]) + (8*(1 + Sqrt[3])*d^3*Sqrt[a + b*x]*(c + d*x)^(1/6))/(27*
b^(5/3)*(b*c - a*d)^2*((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) + (8*d^2*(c + d*x)^(1/6)*((
b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3
) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticE[ArcCos[(
(b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x
)^(1/3))], (2 + Sqrt[3])/4])/(9*3^(3/4)*b^(5/3)*(b*c - a*d)^(5/3)*Sqrt[a + b*x]*Sqrt[-((b^(1/3)*(c + d*x)^(1/3
)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2
)]) + (4*(1 - Sqrt[3])*d^2*(c + d*x)^(1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/
3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b
^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcCos[((b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c
- a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(27*3^(1/4)*b^(5/3)*(b*c - a*d)^(5/3
)*Sqrt[a + b*x]*Sqrt[-((b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1
/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)])

Rule 47

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

Rule 51

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*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 308

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(
(Sqrt[3] - 1)*s^2)/(2*r^2), Int[1/Sqrt[a + b*x^6], x], x] - Dist[1/(2*r^2), Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4
)/Sqrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s
+ r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2
)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1
+ Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, x]

Rule 1881

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/
a, 3]]}, Simp[((1 + Sqrt[3])*d*s^3*x*Sqrt[a + b*x^6])/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2)), x] - Simp[(3^(1/4)*
d*s*x*(s + r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticE[ArcCos[(s + (1 - Sqrt[
3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*
x^2)^2]*Sqrt[a + b*x^6]), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]

Rubi steps

\begin{align*} \int \frac{(c+d x)^{5/6}}{(a+b x)^{7/2}} \, dx &=-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}+\frac{d \int \frac{1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx}{3 b}\\ &=-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac{2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}-\frac{\left (4 d^2\right ) \int \frac{1}{(a+b x)^{3/2} \sqrt [6]{c+d x}} \, dx}{27 b (b c-a d)}\\ &=-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac{2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}+\frac{8 d^2 (c+d x)^{5/6}}{27 b (b c-a d)^2 \sqrt{a+b x}}-\frac{\left (8 d^3\right ) \int \frac{1}{\sqrt{a+b x} \sqrt [6]{c+d x}} \, dx}{81 b (b c-a d)^2}\\ &=-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac{2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}+\frac{8 d^2 (c+d x)^{5/6}}{27 b (b c-a d)^2 \sqrt{a+b x}}-\frac{\left (16 d^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a-\frac{b c}{d}+\frac{b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{27 b (b c-a d)^2}\\ &=-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac{2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}+\frac{8 d^2 (c+d x)^{5/6}}{27 b (b c-a d)^2 \sqrt{a+b x}}+\frac{\left (8 d^2\right ) \operatorname{Subst}\left (\int \frac{\left (-1+\sqrt{3}\right ) (b c-a d)^{2/3}-2 b^{2/3} x^4}{\sqrt{a-\frac{b c}{d}+\frac{b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{27 b^{5/3} (b c-a d)^2}+\frac{\left (8 \left (1-\sqrt{3}\right ) d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-\frac{b c}{d}+\frac{b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{27 b^{5/3} (b c-a d)^{4/3}}\\ &=-\frac{2 (c+d x)^{5/6}}{5 b (a+b x)^{5/2}}-\frac{2 d (c+d x)^{5/6}}{9 b (b c-a d) (a+b x)^{3/2}}+\frac{8 d^2 (c+d x)^{5/6}}{27 b (b c-a d)^2 \sqrt{a+b x}}+\frac{8 \left (1+\sqrt{3}\right ) d^3 \sqrt{a+b x} \sqrt [6]{c+d x}}{27 b^{5/3} (b c-a d)^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac{8 d^2 \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9\ 3^{3/4} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{4 \left (1-\sqrt{3}\right ) d^2 \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{27 \sqrt [4]{3} b^{5/3} (b c-a d)^{5/3} \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0254339, size = 73, normalized size = 0.08 \[ -\frac{2 (c+d x)^{5/6} \, _2F_1\left (-\frac{5}{2},-\frac{5}{6};-\frac{3}{2};\frac{d (a+b x)}{a d-b c}\right )}{5 b (a+b x)^{5/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/6)/(a + b*x)^(7/2),x]

[Out]

(-2*(c + d*x)^(5/6)*Hypergeometric2F1[-5/2, -5/6, -3/2, (d*(a + b*x))/(-(b*c) + a*d)])/(5*b*(a + b*x)^(5/2)*((
b*(c + d*x))/(b*c - a*d))^(5/6))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{{\frac{5}{6}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/6)/(b*x+a)^(7/2),x)

[Out]

int((d*x+c)^(5/6)/(b*x+a)^(7/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b x + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/6)/(b*x+a)^(7/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/6)/(b*x + a)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{5}{6}}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/6)/(b*x+a)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)*(d*x + c)^(5/6)/(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^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((d*x+c)**(5/6)/(b*x+a)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b x + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/6)/(b*x+a)^(7/2),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/6)/(b*x + a)^(7/2), x)