3.18.3 \(\int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx\) [1703]

Optimal. Leaf size=263 \[ \frac {2 (b d-a e)^3 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{7/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

2/3*(-a*e+b*d)^2*(b*x+a)*(e*x+d)^(3/2)/b^3/((b*x+a)^2)^(1/2)+2/5*(-a*e+b*d)*(b*x+a)*(e*x+d)^(5/2)/b^2/((b*x+a)
^2)^(1/2)+2/7*(b*x+a)*(e*x+d)^(7/2)/b/((b*x+a)^2)^(1/2)-2*(-a*e+b*d)^(7/2)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/
2)/(-a*e+b*d)^(1/2))/b^(9/2)/((b*x+a)^2)^(1/2)+2*(-a*e+b*d)^3*(b*x+a)*(e*x+d)^(1/2)/b^4/((b*x+a)^2)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {660, 52, 65, 214} \begin {gather*} \frac {2 (a+b x) (d+e x)^{5/2} (b d-a e)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) \sqrt {d+e x} (b d-a e)^3}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2} (b d-a e)^2}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x])/(b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*(b*d - a*e)^2*(a + b*x)*(d
+ e*x)^(3/2))/(3*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*(b*d - a*e)*(a + b*x)*(d + e*x)^(5/2))/(5*b^2*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) + (2*(a + b*x)*(d + e*x)^(7/2))/(7*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*(b*d - a*e)^(
7/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 52

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

Rule 65

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(d+e x)^{7/2}}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (b d-a e)^3 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right )^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{b^8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (b d-a e)^3 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right )^4 \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^8 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (b d-a e)^3 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{7/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 168, normalized size = 0.64 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x} \left (-105 a^3 e^3+35 a^2 b e^2 (10 d+e x)-7 a b^2 e \left (58 d^2+16 d e x+3 e^2 x^2\right )+b^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )+105 (-b d+a e)^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{105 b^{9/2} \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*(a + b*x)*(Sqrt[b]*Sqrt[d + e*x]*(-105*a^3*e^3 + 35*a^2*b*e^2*(10*d + e*x) - 7*a*b^2*e*(58*d^2 + 16*d*e*x +
3*e^2*x^2) + b^3*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 + 15*e^3*x^3)) + 105*(-(b*d) + a*e)^(7/2)*ArcTan[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/(105*b^(9/2)*Sqrt[(a + b*x)^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(184)=368\).
time = 0.60, size = 462, normalized size = 1.76

method	result	size
risch	\(-\frac {2 \left (-15 b^{3} x^{3} e^{3}+21 a \,b^{2} e^{3} x^{2}-66 b^{3} d \,e^{2} x^{2}-35 a^{2} b \,e^{3} x +112 a \,b^{2} d \,e^{2} x -122 b^{3} d^{2} e x +105 e^{3} a^{3}-350 a^{2} b d \,e^{2}+406 a \,b^{2} d^{2} e -176 b^{3} d^{3}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{105 b^{4} \left (b x +a \right )}+\frac {\left (\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) e^{4} a^{4}}{b^{4} \sqrt {b \left (a e -b d \right )}}-\frac {8 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} d \,e^{3}}{b^{3} \sqrt {b \left (a e -b d \right )}}+\frac {12 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} d^{2} e^{2}}{b^{2} \sqrt {b \left (a e -b d \right )}}-\frac {8 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,d^{3} e}{b \sqrt {b \left (a e -b d \right )}}+\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d^{4}}{\sqrt {b \left (a e -b d \right )}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\)	\(373\)
default	\(\frac {2 \left (b x +a \right ) \left (15 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {7}{2}} b^{3}-21 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} a \,b^{2} e +21 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} b^{3} d +35 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b \,e^{2}-70 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} d e +35 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d^{2}+105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4} e^{4}-420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b d \,e^{3}+630 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{2} d^{2} e^{2}-420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{3} d^{3} e +105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{4} d^{4}-105 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{3} e^{3}+315 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} b d \,e^{2}-315 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{2} d^{2} e +105 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{3} d^{3}\right )}{105 \sqrt {\left (b x +a \right )^{2}}\, b^{4} \sqrt {b \left (a e -b d \right )}}\)	\(462\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/((b*x+a)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/105*(b*x+a)*(15*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^3-21*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^2*e+21*(b*(a*
e-b*d))^(1/2)*(e*x+d)^(5/2)*b^3*d+35*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-70*(b*(a*e-b*d))^(1/2)*(e*x+d
)^(3/2)*a*b^2*d*e+35*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^3*d^2+105*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))
*a^4*e^4-420*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^3*b*d*e^3+630*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^
(1/2))*a^2*b^2*d^2*e^2-420*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a*b^3*d^3*e+105*arctan(b*(e*x+d)^(1/2)/
(b*(a*e-b*d))^(1/2))*b^4*d^4-105*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*e^3+315*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/
2)*a^2*b*d*e^2-315*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d^2*e+105*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d^3
)/((b*x+a)^2)^(1/2)/b^4/(b*(a*e-b*d))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x*e + d)^(7/2)/sqrt((b*x + a)^2), x)

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Fricas [A]
time = 2.53, size = 415, normalized size = 1.58 \begin {gather*} \left [-\frac {105 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d + 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (176 \, b^{3} d^{3} + {\left (15 \, b^{3} x^{3} - 21 \, a b^{2} x^{2} + 35 \, a^{2} b x - 105 \, a^{3}\right )} e^{3} + 2 \, {\left (33 \, b^{3} d x^{2} - 56 \, a b^{2} d x + 175 \, a^{2} b d\right )} e^{2} + 2 \, {\left (61 \, b^{3} d^{2} x - 203 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{105 \, b^{4}}, -\frac {2 \, {\left (105 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (176 \, b^{3} d^{3} + {\left (15 \, b^{3} x^{3} - 21 \, a b^{2} x^{2} + 35 \, a^{2} b x - 105 \, a^{3}\right )} e^{3} + 2 \, {\left (33 \, b^{3} d x^{2} - 56 \, a b^{2} d x + 175 \, a^{2} b d\right )} e^{2} + 2 \, {\left (61 \, b^{3} d^{2} x - 203 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}\right )}}{105 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/105*(105*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt((b*d - a*e)/b)*log((2*b*d + 2*sqrt(x*e +
d)*b*sqrt((b*d - a*e)/b) + (b*x - a)*e)/(b*x + a)) - 2*(176*b^3*d^3 + (15*b^3*x^3 - 21*a*b^2*x^2 + 35*a^2*b*x
- 105*a^3)*e^3 + 2*(33*b^3*d*x^2 - 56*a*b^2*d*x + 175*a^2*b*d)*e^2 + 2*(61*b^3*d^2*x - 203*a*b^2*d^2)*e)*sqrt
(x*e + d))/b^4, -2/105*(105*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-(b*d - a*e)/b)*arctan(-s
qrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (176*b^3*d^3 + (15*b^3*x^3 - 21*a*b^2*x^2 + 35*a^2*b*x - 10
5*a^3)*e^3 + 2*(33*b^3*d*x^2 - 56*a*b^2*d*x + 175*a^2*b*d)*e^2 + 2*(61*b^3*d^2*x - 203*a*b^2*d^2)*e)*sqrt(x*e
+ d))/b^4]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/((b*x+a)**2)**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 deep

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Giac [A]
time = 1.73, size = 354, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) + 105 \, \sqrt {x e + d} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} e \mathrm {sgn}\left (b x + a\right ) - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d e \mathrm {sgn}\left (b x + a\right ) - 315 \, \sqrt {x e + d} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {x e + d} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 105 \, \sqrt {x e + d} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

2*(b^4*d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b*x + a) + 6*a^2*b^2*d^2*e^2*sgn(b*x + a) - 4*a^3*b*d*e^3*sgn(b*x
+ a) + a^4*e^4*sgn(b*x + a))*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) + 2/105*(
15*(x*e + d)^(7/2)*b^6*sgn(b*x + a) + 21*(x*e + d)^(5/2)*b^6*d*sgn(b*x + a) + 35*(x*e + d)^(3/2)*b^6*d^2*sgn(b
*x + a) + 105*sqrt(x*e + d)*b^6*d^3*sgn(b*x + a) - 21*(x*e + d)^(5/2)*a*b^5*e*sgn(b*x + a) - 70*(x*e + d)^(3/2
)*a*b^5*d*e*sgn(b*x + a) - 315*sqrt(x*e + d)*a*b^5*d^2*e*sgn(b*x + a) + 35*(x*e + d)^(3/2)*a^2*b^4*e^2*sgn(b*x
+ a) + 315*sqrt(x*e + d)*a^2*b^4*d*e^2*sgn(b*x + a) - 105*sqrt(x*e + d)*a^3*b^3*e^3*sgn(b*x + a))/b^7

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{7/2}}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/((a + b*x)^2)^(1/2),x)

[Out]

int((d + e*x)^(7/2)/((a + b*x)^2)^(1/2), x)

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