3.22.30 \(\int \frac {(a+b x) (d+e x)^{7/2}}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\) [2130]

Optimal. Leaf size=250 \[ \frac {7 e (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (b d-a e)^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

7/3*e*(-a*e+b*d)*(b*x+a)*(e*x+d)^(3/2)/b^3/((b*x+a)^2)^(1/2)+7/5*e*(b*x+a)*(e*x+d)^(5/2)/b^2/((b*x+a)^2)^(1/2)
-(e*x+d)^(7/2)/b/((b*x+a)^2)^(1/2)-7*e*(-a*e+b*d)^(5/2)*(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)+7*e*(-a*e+b*d)^2*(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 = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {782, 660, 52, 65, 214} \begin {gather*} -\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (a+b x) (b d-a e)^{5/2} \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 {7 e (a+b x) \sqrt {d+e x} (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{3/2} (b d-a e)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

Rule 782

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Dist[e*g*(m/(2*c*(p + 1))), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx}{2 b}\\ &=-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{2 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 e \left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{2 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 \left (b^2 d-a b e\right )^3 \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^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{7/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (b d-a e)^{5/2} (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 = 0.10, size = 180, normalized size = 0.72 \begin {gather*} \frac {e (a+b x) \left (\frac {\sqrt {b} \sqrt {d+e x} \left (105 a^3 e^3+35 a^2 b e^2 (-7 d+2 e x)-7 a b^2 e \left (-23 d^2+24 d e x+2 e^2 x^2\right )+b^3 \left (-15 d^3+116 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )}{e (a+b x)}-105 (-b d+a e)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{15 b^{9/2} \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(e*(a + b*x)*((Sqrt[b]*Sqrt[d + e*x]*(105*a^3*e^3 + 35*a^2*b*e^2*(-7*d + 2*e*x) - 7*a*b^2*e*(-23*d^2 + 24*d*e*
x + 2*e^2*x^2) + b^3*(-15*d^3 + 116*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)))/(e*(a + b*x)) - 105*(-(b*d) + a*e)^(
5/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/(15*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. \(661\) vs. \(2(173)=346\).
time = 0.06, size = 662, normalized size = 2.65

method	result	size
risch	\(\frac {2 e \left (3 b^{2} e^{2} x^{2}-10 a b \,e^{2} x +16 b^{2} d e x +45 a^{2} e^{2}-100 a b d e +58 b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{15 b^{4} \left (b x +a \right )}+\frac {\left (\frac {e^{4} \sqrt {e x +d}\, a^{3}}{b^{4} \left (b e x +a e \right )}-\frac {3 e^{3} \sqrt {e x +d}\, a^{2} d}{b^{3} \left (b e x +a e \right )}+\frac {3 e^{2} \sqrt {e x +d}\, a \,d^{2}}{b^{2} \left (b e x +a e \right )}-\frac {e \sqrt {e x +d}\, d^{3}}{b \left (b e x +a e \right )}-\frac {7 e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3}}{b^{4} \sqrt {\left (a e -b d \right ) b}}+\frac {21 e^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} d}{b^{3} \sqrt {\left (a e -b d \right ) b}}-\frac {21 e^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,d^{2}}{b^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {7 e \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) d^{3}}{b \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\)	\(390\)
default	\(\frac {\left (6 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3} e x +6 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e -20 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e^{2} x +20 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3} d e x -105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b \,e^{4} x +315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{2} d \,e^{3} x -315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{3} d^{2} e^{2} x +105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{4} d^{3} e x -20 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{2}+20 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d e +90 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{3} x -180 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d \,e^{2} x +90 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{2} e x -105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} e^{4}+315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b d \,e^{3}-315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{2} d^{2} e^{2}+105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{3} d^{3} e +105 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{3} e^{3}-225 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b d \,e^{2}+135 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d^{2} e -15 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{3}\right ) \left (b x +a \right )^{2}}{15 \sqrt {\left (a e -b d \right ) b}\, b^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\)	\(662\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/15*(6*(e*x+d)^(5/2)*((a*e-b*d)*b)^(1/2)*b^3*e*x+6*(e*x+d)^(5/2)*((a*e-b*d)*b)^(1/2)*a*b^2*e-20*(e*x+d)^(3/2)
*((a*e-b*d)*b)^(1/2)*a*b^2*e^2*x+20*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*b^3*d*e*x-105*arctan(b*(e*x+d)^(1/2)/((a
*e-b*d)*b)^(1/2))*a^3*b*e^4*x+315*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^2*b^2*d*e^3*x-315*arctan(b*(e*
x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a*b^3*d^2*e^2*x+105*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^4*d^3*e*x-20
*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*a^2*b*e^2+20*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*a*b^2*d*e+90*(e*x+d)^(1/2)*(
(a*e-b*d)*b)^(1/2)*a^2*b*e^3*x-180*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a*b^2*d*e^2*x+90*(e*x+d)^(1/2)*((a*e-b*d)
*b)^(1/2)*b^3*d^2*e*x-105*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^4*e^4+315*arctan(b*(e*x+d)^(1/2)/((a*e
-b*d)*b)^(1/2))*a^3*b*d*e^3-315*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^2*b^2*d^2*e^2+105*arctan(b*(e*x+
d)^(1/2)/((a*e-b*d)*b)^(1/2))*a*b^3*d^3*e+105*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a^3*e^3-225*(e*x+d)^(1/2)*((a*
e-b*d)*b)^(1/2)*a^2*b*d*e^2+135*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a*b^2*d^2*e-15*(e*x+d)^(1/2)*((a*e-b*d)*b)^(
1/2)*b^3*d^3)*(b*x+a)^2/((a*e-b*d)*b)^(1/2)/b^4/((b*x+a)^2)^(3/2)

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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((b*x+a)*(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x + a)*(x*e + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2), x)

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Fricas [A]
time = 2.82, size = 470, normalized size = 1.88 \begin {gather*} \left [\frac {105 \, {\left ({\left (a^{2} b x + a^{3}\right )} e^{3} - 2 \, {\left (a b^{2} d x + a^{2} b d\right )} e^{2} + {\left (b^{3} d^{2} x + a b^{2} d^{2}\right )} e\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 (15 \, b^{3} d^{3} - {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} + 70 \, a^{2} b x + 105 \, a^{3}\right )} e^{3} - {\left (32 \, b^{3} d x^{2} - 168 \, a b^{2} d x - 245 \, a^{2} b d\right )} e^{2} - {\left (116 \, b^{3} d^{2} x + 161 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{30 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {105 \, {\left ({\left (a^{2} b x + a^{3}\right )} e^{3} - 2 \, {\left (a b^{2} d x + a^{2} b d\right )} e^{2} + {\left (b^{3} d^{2} x + a b^{2} d^{2}\right )} e\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 (15 \, b^{3} d^{3} - {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} + 70 \, a^{2} b x + 105 \, a^{3}\right )} e^{3} - {\left (32 \, b^{3} d x^{2} - 168 \, a b^{2} d x - 245 \, a^{2} b d\right )} e^{2} - {\left (116 \, b^{3} d^{2} x + 161 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

[1/30*(105*((a^2*b*x + a^3)*e^3 - 2*(a*b^2*d*x + a^2*b*d)*e^2 + (b^3*d^2*x + a*b^2*d^2)*e)*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*(15*b^3*d^3 - (6*b^3*x^3 - 1
4*a*b^2*x^2 + 70*a^2*b*x + 105*a^3)*e^3 - (32*b^3*d*x^2 - 168*a*b^2*d*x - 245*a^2*b*d)*e^2 - (116*b^3*d^2*x +
161*a*b^2*d^2)*e)*sqrt(x*e + d))/(b^5*x + a*b^4), -1/15*(105*((a^2*b*x + a^3)*e^3 - 2*(a*b^2*d*x + a^2*b*d)*e^
2 + (b^3*d^2*x + a*b^2*d^2)*e)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e))
+ (15*b^3*d^3 - (6*b^3*x^3 - 14*a*b^2*x^2 + 70*a^2*b*x + 105*a^3)*e^3 - (32*b^3*d*x^2 - 168*a*b^2*d*x - 245*a^
2*b*d)*e^2 - (116*b^3*d^2*x + 161*a*b^2*d^2)*e)*sqrt(x*e + d))/(b^5*x + a*b^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right ) \left (d + e x\right )^{\frac {7}{2}}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**(7/2)/((a + b*x)**2)**(3/2), x)

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Giac [A]
time = 2.67, size = 305, normalized size = 1.22 \begin {gather*} \frac {7 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\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} \mathrm {sgn}\left (b x + a\right )} - \frac {\sqrt {x e + d} b^{3} d^{3} e - 3 \, \sqrt {x e + d} a b^{2} d^{2} e^{2} + 3 \, \sqrt {x e + d} a^{2} b d e^{3} - \sqrt {x e + d} a^{3} e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{4} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{8} e + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{8} d e + 45 \, \sqrt {x e + d} b^{8} d^{2} e - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{7} e^{2} - 90 \, \sqrt {x e + d} a b^{7} d e^{2} + 45 \, \sqrt {x e + d} a^{2} b^{6} e^{3}\right )}}{15 \, b^{10} \mathrm {sgn}\left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

7*(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-
b^2*d + a*b*e)*b^4*sgn(b*x + a)) - (sqrt(x*e + d)*b^3*d^3*e - 3*sqrt(x*e + d)*a*b^2*d^2*e^2 + 3*sqrt(x*e + d)*
a^2*b*d*e^3 - sqrt(x*e + d)*a^3*e^4)/(((x*e + d)*b - b*d + a*e)*b^4*sgn(b*x + a)) + 2/15*(3*(x*e + d)^(5/2)*b^
8*e + 10*(x*e + d)^(3/2)*b^8*d*e + 45*sqrt(x*e + d)*b^8*d^2*e - 10*(x*e + d)^(3/2)*a*b^7*e^2 - 90*sqrt(x*e + d
)*a*b^7*d*e^2 + 45*sqrt(x*e + d)*a^2*b^6*e^3)/(b^10*sgn(b*x + a))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(d + e*x)^(7/2))/(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int(((a + b*x)*(d + e*x)^(7/2))/(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)

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