3.4.57 \(\int \frac {(d+e x)^{7/2}}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=248 \[ -\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {d^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}}+\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-11 b e)\right )}{4 b^4 c \left (b x+c x^2\right )} \]

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Rubi [A]  time = 0.39, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {738, 818, 826, 1166, 208} \begin {gather*} \frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-11 b e)\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {d^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(2*b^2*(b*x + c*x^2)^2) + (Sqrt[d + e*x]*(b*c*d^2*(12*c*d - 11*b*e)
+ (2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*x))/(4*b^4*c*(b*x + c*x^2)) - (d^(3/2)*(48*c^2*d^2 - 84*b
*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d - b*e)^(3/2)*(48*c^2*d^2 - 12*b*c*d*e - b
^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(3/2))

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 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 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 )^3} \, dx &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d (12 c d-11 b e)+\frac {1}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 (12 c d-11 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e+b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{4} c d^2 \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right )-\frac {1}{4} e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c}\\ &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 (12 c d-11 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e+b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} d e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right )-\frac {1}{4} c d^2 e \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right )-\frac {1}{4} e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-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^4 c}\\ &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 (12 c d-11 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e+b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\left ((c d-b e)^2 \left (48 c^2 d^2-12 b c d e-b^2 e^2\right )\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 )}{4 b^5 c}+\frac {\left (c d^2 \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right )\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 )}{4 b^5}\\ &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 (12 c d-11 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e+b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {d^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (48 c^2 d^2-12 b c d e-b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.48, size = 263, normalized size = 1.06 \begin {gather*} \frac {-\left (d^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {\sqrt {c d-b e} \left (b^3 e^3+11 b^2 c d e^2-60 b c^2 d^2 e+48 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{c^{3/2}}+\frac {b \sqrt {d+e x} \left (b^4 \left (-e^3\right ) x^2+b^3 c \left (-2 d^3-13 d^2 e x+16 d e^2 x^2+e^3 x^3\right )+b^2 c^2 d x \left (8 d^2-55 d e x+10 e^2 x^2\right )+36 b c^3 d^2 x^2 (d-e x)+24 c^4 d^3 x^3\right )}{c x^2 (b+c x)^2}}{4 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*Sqrt[d + e*x]*(-(b^4*e^3*x^2) + 24*c^4*d^3*x^3 + 36*b*c^3*d^2*x^2*(d - e*x) + b^2*c^2*d*x*(8*d^2 - 55*d*e*
x + 10*e^2*x^2) + b^3*c*(-2*d^3 - 13*d^2*e*x + 16*d*e^2*x^2 + e^3*x^3)))/(c*x^2*(b + c*x)^2) - d^(3/2)*(48*c^2
*d^2 - 84*b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (Sqrt[c*d - b*e]*(48*c^3*d^3 - 60*b*c^2*d^2*e
+ 11*b^2*c*d*e^2 + b^3*e^3)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(3/2))/(4*b^5)

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IntegrateAlgebraic [A]  time = 1.53, size = 469, normalized size = 1.89 \begin {gather*} \frac {\left (-35 b^2 d^{3/2} e^2+84 b c d^{5/2} e-48 c^2 d^{7/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {\left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) (b e-c d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{4 b^5 c^{3/2}}-\frac {\sqrt {d+e x} \left (b^4 d^2 e^4-2 b^4 d e^4 (d+e x)+b^4 e^4 (d+e x)^2-26 b^3 c d^3 e^3+42 b^3 c d^2 e^3 (d+e x)-13 b^3 c d e^3 (d+e x)^2-b^3 c e^3 (d+e x)^3+73 b^2 c^2 d^4 e^2-148 b^2 c^2 d^3 e^2 (d+e x)+85 b^2 c^2 d^2 e^2 (d+e x)^2-10 b^2 c^2 d e^2 (d+e x)^3-72 b c^3 d^5 e+180 b c^3 d^4 e (d+e x)-144 b c^3 d^3 e (d+e x)^2+36 b c^3 d^2 e (d+e x)^3+24 c^4 d^6-72 c^4 d^5 (d+e x)+72 c^4 d^4 (d+e x)^2-24 c^4 d^3 (d+e x)^3\right )}{4 b^4 c e x^2 (b e+c (d+e x)-c d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[d + e*x]*(24*c^4*d^6 - 72*b*c^3*d^5*e + 73*b^2*c^2*d^4*e^2 - 26*b^3*c*d^3*e^3 + b^4*d^2*e^4 - 72*c^
4*d^5*(d + e*x) + 180*b*c^3*d^4*e*(d + e*x) - 148*b^2*c^2*d^3*e^2*(d + e*x) + 42*b^3*c*d^2*e^3*(d + e*x) - 2*b
^4*d*e^4*(d + e*x) + 72*c^4*d^4*(d + e*x)^2 - 144*b*c^3*d^3*e*(d + e*x)^2 + 85*b^2*c^2*d^2*e^2*(d + e*x)^2 - 1
3*b^3*c*d*e^3*(d + e*x)^2 + b^4*e^4*(d + e*x)^2 - 24*c^4*d^3*(d + e*x)^3 + 36*b*c^3*d^2*e*(d + e*x)^3 - 10*b^2
*c^2*d*e^2*(d + e*x)^3 - b^3*c*e^3*(d + e*x)^3))/(b^4*c*e*x^2*(-(c*d) + b*e + c*(d + e*x))^2) + ((-(c*d) + b*e
)^(3/2)*(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4
*b^5*c^(3/2)) + ((-48*c^2*d^(7/2) + 84*b*c*d^(5/2)*e - 35*b^2*d^(3/2)*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*
b^5)

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fricas [B]  time = 0.87, size = 2027, normalized size = 8.17

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)^3,x, algorithm="fricas")

[Out]

[1/8*(((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e
+ 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqr
t((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)) + ((48*c^5*d^3 -
84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2
*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2
*b^4*c*d^3 - (24*b*c^4*d^3 - 36*b^2*c^3*d^2*e + 10*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 - (36*b^2*c^3*d^3 - 55*b^3*c
^2*d^2*e + 16*b^4*c*d*e^2 - b^5*e^3)*x^2 - (8*b^3*c^2*d^3 - 13*b^4*c*d^2*e)*x)*sqrt(e*x + d))/(b^5*c^3*x^4 + 2
*b^6*c^2*x^3 + b^7*c*x^2), 1/8*(2*((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*
b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4
*c*d*e^2 + b^5*e^3)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + ((48
*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)*x^3
+ (48*b^2*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)
/x) - 2*(2*b^4*c*d^3 - (24*b*c^4*d^3 - 36*b^2*c^3*d^2*e + 10*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 - (36*b^2*c^3*d^3
- 55*b^3*c^2*d^2*e + 16*b^4*c*d*e^2 - b^5*e^3)*x^2 - (8*b^3*c^2*d^3 - 13*b^4*c*d^2*e)*x)*sqrt(e*x + d))/(b^5*c
^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), 1/8*(2*((48*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*c^
4*d^3 - 84*b^2*c^3*d^2*e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d*e^2)*x^2)*s
qrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + ((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4
+ 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e
+ 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*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^4*c*d^3 - (24*b*c^4*d^3 - 36*b^2*c^3*d^2*e + 10*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 -
(36*b^2*c^3*d^3 - 55*b^3*c^2*d^2*e + 16*b^4*c*d*e^2 - b^5*e^3)*x^2 - (8*b^3*c^2*d^3 - 13*b^4*c*d^2*e)*x)*sqrt
(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), 1/4*(((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 +
b^3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 -
60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*
e)/c)/(c*d - b*e)) + ((48*c^5*d^3 - 84*b*c^4*d^2*e + 35*b^2*c^3*d*e^2)*x^4 + 2*(48*b*c^4*d^3 - 84*b^2*c^3*d^2*
e + 35*b^3*c^2*d*e^2)*x^3 + (48*b^2*c^3*d^3 - 84*b^3*c^2*d^2*e + 35*b^4*c*d*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x
+ d)*sqrt(-d)/d) - (2*b^4*c*d^3 - (24*b*c^4*d^3 - 36*b^2*c^3*d^2*e + 10*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 - (36*
b^2*c^3*d^3 - 55*b^3*c^2*d^2*e + 16*b^4*c*d*e^2 - b^5*e^3)*x^2 - (8*b^3*c^2*d^3 - 13*b^4*c*d^2*e)*x)*sqrt(e*x
+ d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2)]

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giac [B]  time = 0.28, size = 552, normalized size = 2.23 \begin {gather*} \frac {{\left (48 \, c^{2} d^{4} - 84 \, b c d^{3} e + 35 \, b^{2} d^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} - \frac {{\left (48 \, c^{4} d^{4} - 108 \, b c^{3} d^{3} e + 71 \, b^{2} c^{2} d^{2} e^{2} - 10 \, b^{3} c d e^{3} - b^{4} e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, \sqrt {-c^{2} d + b c e} b^{5} c} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{4} d^{3} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{4} d^{4} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{5} e - 24 \, \sqrt {x e + d} c^{4} d^{6} e - 36 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{3} d^{2} e^{2} + 144 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{3} d^{3} e^{2} - 180 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{3} d^{4} e^{2} + 72 \, \sqrt {x e + d} b c^{3} d^{5} e^{2} + 10 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{2} d e^{3} - 85 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{2} d^{2} e^{3} + 148 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{3} e^{3} - 73 \, \sqrt {x e + d} b^{2} c^{2} d^{4} e^{3} + {\left (x e + d\right )}^{\frac {7}{2}} b^{3} c e^{4} + 13 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c d e^{4} - 42 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c d^{2} e^{4} + 26 \, \sqrt {x e + d} b^{3} c d^{3} e^{4} - {\left (x e + d\right )}^{\frac {5}{2}} b^{4} e^{5} + 2 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d e^{5} - \sqrt {x e + d} b^{4} d^{2} e^{5}}{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 )}^{2} b^{4} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(48*c^2*d^4 - 84*b*c*d^3*e + 35*b^2*d^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) - 1/4*(48*c^4*d
^4 - 108*b*c^3*d^3*e + 71*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e^3 - b^4*e^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c
*e))/(sqrt(-c^2*d + b*c*e)*b^5*c) + 1/4*(24*(x*e + d)^(7/2)*c^4*d^3*e - 72*(x*e + d)^(5/2)*c^4*d^4*e + 72*(x*e
+ d)^(3/2)*c^4*d^5*e - 24*sqrt(x*e + d)*c^4*d^6*e - 36*(x*e + d)^(7/2)*b*c^3*d^2*e^2 + 144*(x*e + d)^(5/2)*b*
c^3*d^3*e^2 - 180*(x*e + d)^(3/2)*b*c^3*d^4*e^2 + 72*sqrt(x*e + d)*b*c^3*d^5*e^2 + 10*(x*e + d)^(7/2)*b^2*c^2*
d*e^3 - 85*(x*e + d)^(5/2)*b^2*c^2*d^2*e^3 + 148*(x*e + d)^(3/2)*b^2*c^2*d^3*e^3 - 73*sqrt(x*e + d)*b^2*c^2*d^
4*e^3 + (x*e + d)^(7/2)*b^3*c*e^4 + 13*(x*e + d)^(5/2)*b^3*c*d*e^4 - 42*(x*e + d)^(3/2)*b^3*c*d^2*e^4 + 26*sqr
t(x*e + d)*b^3*c*d^3*e^4 - (x*e + d)^(5/2)*b^4*e^5 + 2*(x*e + d)^(3/2)*b^4*d*e^5 - sqrt(x*e + d)*b^4*d^2*e^5)/
(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c)

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maple [B]  time = 0.06, size = 627, normalized size = 2.53 \begin {gather*} \frac {15 \sqrt {e x +d}\, d \,e^{4}}{4 \left (c e x +b e \right )^{2} b}-\frac {39 \sqrt {e x +d}\, c \,d^{2} e^{3}}{4 \left (c e x +b e \right )^{2} b^{2}}+\frac {37 \sqrt {e x +d}\, c^{2} d^{3} e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {3 \sqrt {e x +d}\, c^{3} d^{4} e}{\left (c e x +b e \right )^{2} b^{4}}-\frac {\sqrt {e x +d}\, e^{5}}{4 \left (c e x +b e \right )^{2} c}+\frac {e^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b c}+\frac {\left (e x +d \right )^{\frac {3}{2}} e^{4}}{4 \left (c e x +b e \right )^{2} b}+\frac {5 \left (e x +d \right )^{\frac {3}{2}} c d \,e^{3}}{2 \left (c e x +b e \right )^{2} b^{2}}+\frac {5 d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {23 \left (e x +d \right )^{\frac {3}{2}} c^{2} d^{2} e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {71 c \,d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c^{3} d^{3} e}{\left (c e x +b e \right )^{2} b^{4}}+\frac {27 c^{2} 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^{4}}-\frac {12 c^{3} 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^{5}}-\frac {35 d^{\frac {3}{2}} e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3}}+\frac {21 c \,d^{\frac {5}{2}} e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4}}-\frac {12 c^{2} d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5}}+\frac {11 \sqrt {e x +d}\, d^{3}}{4 b^{3} x^{2}}-\frac {3 \sqrt {e x +d}\, c \,d^{4}}{b^{4} e \,x^{2}}-\frac {13 \left (e x +d \right )^{\frac {3}{2}} d^{2}}{4 b^{3} x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c \,d^{3}}{b^{4} e \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)+5/2*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*c*d-23/4*e^2/b^3/(c*e*x+b*e)^2*(
e*x+d)^(3/2)*c^2*d^2+3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*c^3*d^3-1/4*e^5/(c*e*x+b*e)^2/c*(e*x+d)^(1/2)+15/4*e^
4/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d-39/4*e^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*d^2+37/4*e^2/b^3/(c*e*x+b*e)^2*(e
*x+d)^(1/2)*d^3*c^2-3*e/b^4/(c*e*x+b*e)^2*c^3*(e*x+d)^(1/2)*d^4+1/4*e^4/b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)
^(1/2)/((b*e-c*d)*c)^(1/2)*c)+5/2*e^3/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d-71
/4*e^2/b^3*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d^2+27*e/b^4/((b*e-c*d)*c)^(1/2)*
arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d^3*c^2-12/b^5*c^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-
c*d)*c)^(1/2)*c)*d^4-13/4*d^2/b^3/x^2*(e*x+d)^(3/2)+3/e*d^3/b^4/x^2*(e*x+d)^(3/2)*c+11/4*d^3/b^3/x^2*(e*x+d)^(
1/2)-3/e*d^4/b^4/x^2*c*(e*x+d)^(1/2)-35/4*e^2*d^(3/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))+21*e*d^(5/2)/b^4*arct
anh((e*x+d)^(1/2)/d^(1/2))*c-12*d^(7/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*c^2

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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)^3,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 = 0.73, size = 1792, normalized size = 7.23

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)^3,x)

[Out]

- (((d + e*x)^(1/2)*(24*c^4*d^6*e + b^4*d^2*e^5 - 72*b*c^3*d^5*e^2 - 26*b^3*c*d^3*e^4 + 73*b^2*c^2*d^4*e^3))/(
4*b^4*c) - (e*(d + e*x)^(7/2)*(b^3*e^3 + 24*c^3*d^3 - 36*b*c^2*d^2*e + 10*b^2*c*d*e^2))/(4*b^4) - ((d + e*x)^(
3/2)*(b^4*d*e^5 + 36*c^4*d^5*e - 90*b*c^3*d^4*e^2 - 21*b^3*c*d^2*e^4 + 74*b^2*c^2*d^3*e^3))/(2*b^4*c) + (e*(d
+ e*x)^(5/2)*(b^4*e^4 + 72*c^4*d^4 + 85*b^2*c^2*d^2*e^2 - 144*b*c^3*d^3*e - 13*b^3*c*d*e^3))/(4*b^4*c))/(c^2*(
d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2
*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) - (atanh((35*e^12*(d^3)^(1/2)*(d + e
*x)^(1/2))/(32*((35*d^2*e^12)/32 + (77*c*d^3*e^11)/(4*b) - (1551*c^2*d^4*e^10)/(16*b^2) + (5223*c^3*d^5*e^9)/(
32*b^3) - (945*c^4*d^6*e^8)/(8*b^4) + (63*c^5*d^7*e^7)/(2*b^5))) + (77*d*e^11*(d^3)^(1/2)*(d + e*x)^(1/2))/(4*
((77*d^3*e^11)/4 + (35*b*d^2*e^12)/(32*c) - (1551*c*d^4*e^10)/(16*b) + (5223*c^2*d^5*e^9)/(32*b^2) - (945*c^3*
d^6*e^8)/(8*b^3) + (63*c^4*d^7*e^7)/(2*b^4))) + (5223*c^2*d^3*e^9*(d^3)^(1/2)*(d + e*x)^(1/2))/(32*((77*b^2*d^
3*e^11)/4 + (5223*c^2*d^5*e^9)/32 - (945*c^3*d^6*e^8)/(8*b) + (35*b^3*d^2*e^12)/(32*c) + (63*c^4*d^7*e^7)/(2*b
^2) - (1551*b*c*d^4*e^10)/16)) - (945*c^3*d^4*e^8*(d^3)^(1/2)*(d + e*x)^(1/2))/(8*((77*b^3*d^3*e^11)/4 - (945*
c^3*d^6*e^8)/8 + (5223*b*c^2*d^5*e^9)/32 - (1551*b^2*c*d^4*e^10)/16 + (63*c^4*d^7*e^7)/(2*b) + (35*b^4*d^2*e^1
2)/(32*c))) + (63*c^4*d^5*e^7*(d^3)^(1/2)*(d + e*x)^(1/2))/(2*((77*b^4*d^3*e^11)/4 + (63*c^4*d^7*e^7)/2 - (945
*b*c^3*d^6*e^8)/8 - (1551*b^3*c*d^4*e^10)/16 + (5223*b^2*c^2*d^5*e^9)/32 + (35*b^5*d^2*e^12)/(32*c))) - (1551*
c*d^2*e^10*(d^3)^(1/2)*(d + e*x)^(1/2))/(16*((77*b*d^3*e^11)/4 - (1551*c*d^4*e^10)/16 + (5223*c^2*d^5*e^9)/(32
*b) + (35*b^2*d^2*e^12)/(32*c) - (945*c^3*d^6*e^8)/(8*b^2) + (63*c^4*d^7*e^7)/(2*b^3))))*(d^3)^(1/2)*(35*b^2*e
^2 + 48*c^2*d^2 - 84*b*c*d*e))/(4*b^5) - (atanh((183*d^3*e^9*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^
4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(32*((31*b^3*d^2*e^12)/32 + (3711*c^3*d^5*e^9)/32 - (1593*b*c^2*d^4*e^10)/32 +
(59*b^2*c*d^3*e^11)/16 + (b^4*d*e^13)/(32*c) - (819*c^4*d^6*e^8)/(8*b) + (63*c^5*d^7*e^7)/(2*b^2))) - (315*d^
4*e^8*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(8*((59*b^3*d^3*e^11)/1
6 - (819*c^3*d^6*e^8)/8 + (3711*b*c^2*d^5*e^9)/32 - (1593*b^2*c*d^4*e^10)/32 + (b^5*d*e^13)/(32*c^2) + (63*c^4
*d^7*e^7)/(2*b) + (31*b^4*d^2*e^12)/(32*c))) + (33*d^2*e^10*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^4
*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(32*((b^3*d*e^13)/32 - (1593*c^3*d^4*e^10)/32 + (59*b*c^2*d^3*e^11)/16 + (31*b^
2*c*d^2*e^12)/32 + (3711*c^4*d^5*e^9)/(32*b) - (819*c^5*d^6*e^8)/(8*b^2) + (63*c^6*d^7*e^7)/(2*b^3))) + (d*e^1
1*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(32*((59*c^3*d^3*e^11)/16 +
(31*b*c^2*d^2*e^12)/32 - (1593*c^4*d^4*e^10)/(32*b) + (3711*c^5*d^5*e^9)/(32*b^2) - (819*c^6*d^6*e^8)/(8*b^3)
+ (63*c^7*d^7*e^7)/(2*b^4) + (b^2*c*d*e^13)/32)) + (63*c*d^5*e^7*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b
^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(2*((59*b^4*d^3*e^11)/16 + (63*c^4*d^7*e^7)/2 - (819*b*c^3*d^6*e^8)/8 - (
1593*b^3*c*d^4*e^10)/32 + (b^6*d*e^13)/(32*c^2) + (3711*b^2*c^2*d^5*e^9)/32 + (31*b^5*d^2*e^12)/(32*c))))*(-c^
3*(b*e - c*d)^3)^(1/2)*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e))/(4*b^5*c^3)

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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)**3,x)

[Out]

Timed out

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