3.4.48 \(\int \frac {(d+e x)^{7/2}}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=200 \[ -\frac {(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{5/2}}+\frac {d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {e \sqrt {d+e x} \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 c^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac {e (d+e x)^{3/2} (2 c d-b e)}{b^2 c} \]

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Rubi [A]  time = 0.40, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {738, 824, 826, 1166, 208} \begin {gather*} \frac {e \sqrt {d+e x} \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 c^2}-\frac {(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{5/2}}+\frac {d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac {e (d+e x)^{3/2} (2 c d-b e)}{b^2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 208

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

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

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

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Mathematica [A]  time = 0.31, size = 167, normalized size = 0.84 \begin {gather*} \frac {\frac {b \sqrt {d+e x} \left (3 b^3 e^3 x+b^2 c e^2 x (2 e x-3 d)-b c^2 d^2 (d-3 e x)-2 c^3 d^3 x\right )}{c^2 x (b+c x)}-\frac {(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{c^{5/2}}+d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.74, size = 264, normalized size = 1.32 \begin {gather*} \frac {(3 b e+4 c d) (b e-c d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 c^{5/2}}+\frac {\left (4 c d^{7/2}-7 b d^{5/2} e\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {\sqrt {d+e x} \left (3 b^3 e^4 (d+e x)-3 b^3 d e^4+5 b^2 c d^2 e^3-7 b^2 c d e^3 (d+e x)+2 b^2 c e^3 (d+e x)^2-4 b c^2 d^3 e^2+3 b c^2 d^2 e^2 (d+e x)+2 c^3 d^4 e-2 c^3 d^3 e (d+e x)\right )}{b^2 c^2 e x (b e+c (d+e x)-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(7/2)/(b*x + c*x^2)^2,x]

[Out]

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

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fricas [A]  time = 1.05, size = 1273, normalized size = 6.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^
3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c)
)/(c*x + b)) - ((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sqrt(d)*log((e*x - 2*sqrt
(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^
2 - 3*b^4*e^3)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), -1/2*(2*((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*
e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt(-(c*d - b*e)/c)*a
rctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + ((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7
*b^2*c^2*d^2*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (
2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^2 - 3*b^4*e^3)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), -1/2*(
2*((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)
/d) - ((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^
3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c)
)/(c*x + b)) - 2*(2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^2 - 3*b^4*e^3)*
x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), -(((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^
2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*
c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + ((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sq
rt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (2*b^3*c*e^3*x^2 - b^2*c^2*d^3 - (2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*
b^3*c*d*e^2 - 3*b^4*e^3)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x)]

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giac [A]  time = 0.23, size = 344, normalized size = 1.72 \begin {gather*} \frac {2 \, \sqrt {x e + d} e^{3}}{c^{2}} - \frac {{\left (4 \, c d^{4} - 7 \, b d^{3} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} + \frac {{\left (4 \, c^{4} d^{4} - 9 \, b c^{3} d^{3} e + 3 \, b^{2} c^{2} d^{2} e^{2} + 5 \, b^{3} c d e^{3} - 3 \, b^{4} e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{2}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e - 2 \, \sqrt {x e + d} c^{3} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{2} + 4 \, \sqrt {x e + d} b c^{2} d^{3} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{3} - 3 \, \sqrt {x e + d} b^{2} c d^{2} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{4} + \sqrt {x e + d} b^{3} d e^{4}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )} b^{2} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x*e + d)*e^3/c^2 - (4*c*d^4 - 7*b*d^3*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) + (4*c^4*d^4 - 9
*b*c^3*d^3*e + 3*b^2*c^2*d^2*e^2 + 5*b^3*c*d*e^3 - 3*b^4*e^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sq
rt(-c^2*d + b*c*e)*b^3*c^2) - (2*(x*e + d)^(3/2)*c^3*d^3*e - 2*sqrt(x*e + d)*c^3*d^4*e - 3*(x*e + d)^(3/2)*b*c
^2*d^2*e^2 + 4*sqrt(x*e + d)*b*c^2*d^3*e^2 + 3*(x*e + d)^(3/2)*b^2*c*d*e^3 - 3*sqrt(x*e + d)*b^2*c*d^2*e^3 - (
x*e + d)^(3/2)*b^3*e^4 + sqrt(x*e + d)*b^3*d*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e -
b*d*e)*b^2*c^2)

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maple [B]  time = 0.07, size = 403, normalized size = 2.02 \begin {gather*} -\frac {3 b \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c^{2}}+\frac {3 d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b}-\frac {9 c \,d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {4 c^{2} d^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {5 d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c}+\frac {\sqrt {e x +d}\, b \,e^{4}}{\left (c e x +b e \right ) c^{2}}+\frac {3 \sqrt {e x +d}\, d^{2} e^{2}}{\left (c e x +b e \right ) b}-\frac {\sqrt {e x +d}\, c \,d^{3} e}{\left (c e x +b e \right ) b^{2}}-\frac {3 \sqrt {e x +d}\, d \,e^{3}}{\left (c e x +b e \right ) c}-\frac {7 d^{\frac {5}{2}} e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {4 c \,d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3}}+\frac {2 \sqrt {e x +d}\, e^{3}}{c^{2}}-\frac {\sqrt {e x +d}\, d^{3}}{b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 0.78, size = 2913, normalized size = 14.56

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d + e*x)^(1/2)*(b^3*d*e^4 - 2*c^3*d^4*e + 4*b*c^2*d^3*e^2 - 3*b^2*c*d^2*e^3))/b^2 - ((d + e*x)^(3/2)*(b^3*e
^4 - 2*c^3*d^3*e + 3*b*c^2*d^2*e^2 - 3*b^2*c*d*e^3))/b^2)/((2*c^3*d - b*c^2*e)*(d + e*x) - c^3*(d + e*x)^2 - c
^3*d^2 + b*c^2*d*e) + (2*e^3*(d + e*x)^(1/2))/c^2 + (atan(((((2*(d + e*x)^(1/2)*(9*b^8*e^10 + 32*c^8*d^8*e^2 -
128*b*c^7*d^7*e^3 + 154*b^2*c^6*d^6*e^4 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 + 84*b^5*c^3*d^3*e^7 + 7*b
^6*c^2*d^2*e^8 - 30*b^7*c*d*e^9))/(b^4*c^3) + (((12*b^9*c^3*d*e^6 - 8*b^6*c^6*d^4*e^3 + 16*b^7*c^5*d^3*e^4 - 2
0*b^8*c^4*d^2*e^5)/(b^6*c^3) + ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(7*b*e - 4*c*d)*(d^5)^(1/2)*(d + e*x)^(1/2))
/(b^7*c^3))*(7*b*e - 4*c*d)*(d^5)^(1/2))/(2*b^3))*(7*b*e - 4*c*d)*(d^5)^(1/2)*1i)/(2*b^3) + (((2*(d + e*x)^(1/
2)*(9*b^8*e^10 + 32*c^8*d^8*e^2 - 128*b*c^7*d^7*e^3 + 154*b^2*c^6*d^6*e^4 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d
^4*e^6 + 84*b^5*c^3*d^3*e^7 + 7*b^6*c^2*d^2*e^8 - 30*b^7*c*d*e^9))/(b^4*c^3) - (((12*b^9*c^3*d*e^6 - 8*b^6*c^6
*d^4*e^3 + 16*b^7*c^5*d^3*e^4 - 20*b^8*c^4*d^2*e^5)/(b^6*c^3) - ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(7*b*e - 4*
c*d)*(d^5)^(1/2)*(d + e*x)^(1/2))/(b^7*c^3))*(7*b*e - 4*c*d)*(d^5)^(1/2))/(2*b^3))*(7*b*e - 4*c*d)*(d^5)^(1/2)
*1i)/(2*b^3))/((2*(63*b^8*d^3*e^11 + 32*c^8*d^11*e^3 - 176*b*c^7*d^10*e^4 - 246*b^7*c*d^4*e^10 + 262*b^2*c^6*d
^9*e^5 + 141*b^3*c^5*d^8*e^6 - 658*b^4*c^4*d^7*e^7 + 413*b^5*c^3*d^6*e^8 + 169*b^6*c^2*d^5*e^9))/(b^6*c^3) + (
((2*(d + e*x)^(1/2)*(9*b^8*e^10 + 32*c^8*d^8*e^2 - 128*b*c^7*d^7*e^3 + 154*b^2*c^6*d^6*e^4 - 14*b^3*c^5*d^5*e^
5 - 105*b^4*c^4*d^4*e^6 + 84*b^5*c^3*d^3*e^7 + 7*b^6*c^2*d^2*e^8 - 30*b^7*c*d*e^9))/(b^4*c^3) + (((12*b^9*c^3*
d*e^6 - 8*b^6*c^6*d^4*e^3 + 16*b^7*c^5*d^3*e^4 - 20*b^8*c^4*d^2*e^5)/(b^6*c^3) + ((4*b^7*c^5*e^3 - 8*b^6*c^6*d
*e^2)*(7*b*e - 4*c*d)*(d^5)^(1/2)*(d + e*x)^(1/2))/(b^7*c^3))*(7*b*e - 4*c*d)*(d^5)^(1/2))/(2*b^3))*(7*b*e - 4
*c*d)*(d^5)^(1/2))/(2*b^3) - (((2*(d + e*x)^(1/2)*(9*b^8*e^10 + 32*c^8*d^8*e^2 - 128*b*c^7*d^7*e^3 + 154*b^2*c
^6*d^6*e^4 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 + 84*b^5*c^3*d^3*e^7 + 7*b^6*c^2*d^2*e^8 - 30*b^7*c*d*e^
9))/(b^4*c^3) - (((12*b^9*c^3*d*e^6 - 8*b^6*c^6*d^4*e^3 + 16*b^7*c^5*d^3*e^4 - 20*b^8*c^4*d^2*e^5)/(b^6*c^3) -
((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(7*b*e - 4*c*d)*(d^5)^(1/2)*(d + e*x)^(1/2))/(b^7*c^3))*(7*b*e - 4*c*d)*(d
^5)^(1/2))/(2*b^3))*(7*b*e - 4*c*d)*(d^5)^(1/2))/(2*b^3)))*(7*b*e - 4*c*d)*(d^5)^(1/2)*1i)/b^3 + (atan((((-c^5
*(b*e - c*d)^5)^(1/2)*(3*b*e + 4*c*d)*((2*(d + e*x)^(1/2)*(9*b^8*e^10 + 32*c^8*d^8*e^2 - 128*b*c^7*d^7*e^3 + 1
54*b^2*c^6*d^6*e^4 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 + 84*b^5*c^3*d^3*e^7 + 7*b^6*c^2*d^2*e^8 - 30*b^
7*c*d*e^9))/(b^4*c^3) + ((-c^5*(b*e - c*d)^5)^(1/2)*((12*b^9*c^3*d*e^6 - 8*b^6*c^6*d^4*e^3 + 16*b^7*c^5*d^3*e^
4 - 20*b^8*c^4*d^2*e^5)/(b^6*c^3) + ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(-c^5*(b*e - c*d)^5)^(1/2)*(3*b*e + 4*c
*d)*(d + e*x)^(1/2))/(b^7*c^8))*(3*b*e + 4*c*d))/(2*b^3*c^5))*1i)/(2*b^3*c^5) + ((-c^5*(b*e - c*d)^5)^(1/2)*(3
*b*e + 4*c*d)*((2*(d + e*x)^(1/2)*(9*b^8*e^10 + 32*c^8*d^8*e^2 - 128*b*c^7*d^7*e^3 + 154*b^2*c^6*d^6*e^4 - 14*
b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 + 84*b^5*c^3*d^3*e^7 + 7*b^6*c^2*d^2*e^8 - 30*b^7*c*d*e^9))/(b^4*c^3) -
((-c^5*(b*e - c*d)^5)^(1/2)*((12*b^9*c^3*d*e^6 - 8*b^6*c^6*d^4*e^3 + 16*b^7*c^5*d^3*e^4 - 20*b^8*c^4*d^2*e^5)/
(b^6*c^3) - ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(-c^5*(b*e - c*d)^5)^(1/2)*(3*b*e + 4*c*d)*(d + e*x)^(1/2))/(b^
7*c^8))*(3*b*e + 4*c*d))/(2*b^3*c^5))*1i)/(2*b^3*c^5))/((2*(63*b^8*d^3*e^11 + 32*c^8*d^11*e^3 - 176*b*c^7*d^10
*e^4 - 246*b^7*c*d^4*e^10 + 262*b^2*c^6*d^9*e^5 + 141*b^3*c^5*d^8*e^6 - 658*b^4*c^4*d^7*e^7 + 413*b^5*c^3*d^6*
e^8 + 169*b^6*c^2*d^5*e^9))/(b^6*c^3) + ((-c^5*(b*e - c*d)^5)^(1/2)*(3*b*e + 4*c*d)*((2*(d + e*x)^(1/2)*(9*b^8
*e^10 + 32*c^8*d^8*e^2 - 128*b*c^7*d^7*e^3 + 154*b^2*c^6*d^6*e^4 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 +
84*b^5*c^3*d^3*e^7 + 7*b^6*c^2*d^2*e^8 - 30*b^7*c*d*e^9))/(b^4*c^3) + ((-c^5*(b*e - c*d)^5)^(1/2)*((12*b^9*c^3
*d*e^6 - 8*b^6*c^6*d^4*e^3 + 16*b^7*c^5*d^3*e^4 - 20*b^8*c^4*d^2*e^5)/(b^6*c^3) + ((4*b^7*c^5*e^3 - 8*b^6*c^6*
d*e^2)*(-c^5*(b*e - c*d)^5)^(1/2)*(3*b*e + 4*c*d)*(d + e*x)^(1/2))/(b^7*c^8))*(3*b*e + 4*c*d))/(2*b^3*c^5)))/(
2*b^3*c^5) - ((-c^5*(b*e - c*d)^5)^(1/2)*(3*b*e + 4*c*d)*((2*(d + e*x)^(1/2)*(9*b^8*e^10 + 32*c^8*d^8*e^2 - 12
8*b*c^7*d^7*e^3 + 154*b^2*c^6*d^6*e^4 - 14*b^3*c^5*d^5*e^5 - 105*b^4*c^4*d^4*e^6 + 84*b^5*c^3*d^3*e^7 + 7*b^6*
c^2*d^2*e^8 - 30*b^7*c*d*e^9))/(b^4*c^3) - ((-c^5*(b*e - c*d)^5)^(1/2)*((12*b^9*c^3*d*e^6 - 8*b^6*c^6*d^4*e^3
+ 16*b^7*c^5*d^3*e^4 - 20*b^8*c^4*d^2*e^5)/(b^6*c^3) - ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(-c^5*(b*e - c*d)^5)
^(1/2)*(3*b*e + 4*c*d)*(d + e*x)^(1/2))/(b^7*c^8))*(3*b*e + 4*c*d))/(2*b^3*c^5)))/(2*b^3*c^5)))*(-c^5*(b*e - c
*d)^5)^(1/2)*(3*b*e + 4*c*d)*1i)/(b^3*c^5)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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