3.638 \(\int \frac {(d+e x)^{7/2}}{(a-c x^2)^3} \, dx\)
Optimal. Leaf size=294 \[ -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (-18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{9/4}}+\frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]
[Out]
1/4*(c*d*x+a*e)*(e*x+d)^(5/2)/a/c/(-c*x^2+a)^2+1/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))
*(e*a^(1/2)+d*c^(1/2))^(3/2)*(12*c*d^2+5*a*e^2-18*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(9/4)-1/32*arctanh(c^(1/4)*(e
*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(-e*a^(1/2)+d*c^(1/2))^(3/2)*(12*c*d^2+5*a*e^2+18*d*e*a^(1/2)*c^(1/2
))/a^(5/2)/c^(9/4)+1/16*(a*e*(-5*a*e^2+7*c*d^2)+2*c*d*(-2*a*e^2+3*c*d^2)*x)*(e*x+d)^(1/2)/a^2/c^2/(-c*x^2+a)
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Rubi [A] time = 0.52, antiderivative size = 294, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used =
{739, 819, 827, 1166, 208} \[ \frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (-18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{9/4}}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(7/2)/(a - c*x^2)^3,x]
[Out]
((a*e + c*d*x)*(d + e*x)^(5/2))/(4*a*c*(a - c*x^2)^2) + (Sqrt[d + e*x]*(a*e*(7*c*d^2 - 5*a*e^2) + 2*c*d*(3*c*d
^2 - 2*a*e^2)*x))/(16*a^2*c^2*(a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e
+ 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4)) + ((Sqrt[c]*d +
Sqrt[a]*e)^(3/2)*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d
+ Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))
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 739
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 819
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Rule 827
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + 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 (a-c x^2\right )^3} \, dx &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (-6 c d^2+5 a e^2\right )-\frac {1}{2} c d e x\right )}{\left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{4} \left (4 c d^2-5 a e^2\right ) \left (3 c d^2-a e^2\right )+\frac {1}{2} c d e \left (3 c d^2-4 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2}\\ &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} c d^2 e \left (3 c d^2-4 a e^2\right )+\frac {1}{4} e \left (4 c d^2-5 a e^2\right ) \left (3 c d^2-a e^2\right )+\frac {1}{2} c d e \left (3 c d^2-4 a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{4 a^2 c^2}\\ &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\left (\left (\sqrt {c} d+\sqrt {a} e\right )^2 \left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} c^{3/2}}-\frac {\left (\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} c^{3/2}}\\ &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 319, normalized size = 1.09 \[ \frac {-\sqrt {\sqrt {c} d-\sqrt {a} e} \left (-5 a^{3/2} e^3+6 \sqrt {a} c d^2 e-13 a \sqrt {c} d e^2+12 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )+\sqrt {\sqrt {a} e+\sqrt {c} d} \left (5 a^{3/2} e^3-6 \sqrt {a} c d^2 e-13 a \sqrt {c} d e^2+12 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )+\frac {2 \sqrt {a} \sqrt [4]{c} \sqrt {d+e x} \left (-5 a^3 e^3+a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )-6 c^3 d^3 x^3\right )}{\left (a-c x^2\right )^2}}{32 a^{5/2} c^{9/4}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(7/2)/(a - c*x^2)^3,x]
[Out]
((2*Sqrt[a]*c^(1/4)*Sqrt[d + e*x]*(-5*a^3*e^3 - 6*c^3*d^3*x^3 + a*c^2*d*x*(10*d^2 + d*e*x + 8*e^2*x^2) + a^2*c
*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)))/(a - c*x^2)^2 - Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(12*c^(3/2)*d^3 + 6*Sqrt[a]*c*
d^2*e - 13*a*Sqrt[c]*d*e^2 - 5*a^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + Sqr
t[Sqrt[c]*d + Sqrt[a]*e]*(12*c^(3/2)*d^3 - 6*Sqrt[a]*c*d^2*e - 13*a*Sqrt[c]*d*e^2 + 5*a^(3/2)*e^3)*ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))
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fricas [B] time = 1.24, size = 1730, normalized size = 5.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="fricas")
[Out]
1/64*((a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*
a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((302
4*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt(e*x + d
) + (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^3 - 13*a^6*c^7*d*e^2)*sqrt((
441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a
^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/
(a^5*c^4))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^
4 - 105*a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*
log((3024*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt
(e*x + d) - (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^3 - 13*a^6*c^7*d*e^2
)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2
+ 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5
*c^9)))/(a^5*c^4))) + (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*
c*d^3*e^4 - 105*a^3*d*e^6 - a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^
5*c^4))*log((3024*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 625*a^4*e^
13)*sqrt(e*x + d) + (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 - 13*a^6*c
^7*d*e^2)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - 420*a*c^2
*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^
14)/(a^5*c^9)))/(a^5*c^4))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 +
385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^
9)))/(a^5*c^4))*log((3024*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 62
5*a^4*e^13)*sqrt(e*x + d) - (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 -
13*a^6*c^7*d*e^2)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - 4
20*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 62
5*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))) + 4*(11*a^2*c*d^2*e - 5*a^3*e^3 - 2*(3*c^3*d^3 - 4*a*c^2*d*e^2)*x^3 + (a*c
^2*d^2*e + 9*a^2*c*e^3)*x^2 + 2*(5*a*c^2*d^3 + 2*a^2*c*d*e^2)*x)*sqrt(e*x + d))/(a^2*c^4*x^4 - 2*a^3*c^3*x^2 +
a^4*c^2)
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giac [B] time = 0.56, size = 535, normalized size = 1.82 \[ -\frac {{\left (6 \, a c d^{2} e - 12 \, \sqrt {a c} c d^{3} + 13 \, \sqrt {a c} a d e^{2} - 5 \, a^{2} e^{3}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d + \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, a^{3} c^{4}} - \frac {{\left (6 \, a c d^{2} e + 12 \, \sqrt {a c} c d^{3} - 13 \, \sqrt {a c} a d e^{2} - 5 \, a^{2} e^{3}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d - \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, a^{3} c^{4}} - \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{3} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{4} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{5} e - 6 \, \sqrt {x e + d} c^{3} d^{6} e - 8 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} d e^{3} + 23 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d^{2} e^{3} - 32 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{3} + 17 \, \sqrt {x e + d} a c^{2} d^{4} e^{3} - 9 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} c e^{5} + 14 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c d e^{5} - 16 \, \sqrt {x e + d} a^{2} c d^{2} e^{5} + 5 \, \sqrt {x e + d} a^{3} e^{7}}{16 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="giac")
[Out]
-1/32*(6*a*c*d^2*e - 12*sqrt(a*c)*c*d^3 + 13*sqrt(a*c)*a*d*e^2 - 5*a^2*e^3)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(c
)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d + sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2*e^2)*a^2*c^3))/(a^2*c^3)))
/(a^3*c^4) - 1/32*(6*a*c*d^2*e + 12*sqrt(a*c)*c*d^3 - 13*sqrt(a*c)*a*d*e^2 - 5*a^2*e^3)*sqrt(-c^2*d + sqrt(a*c
)*c*e)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d - sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2*e^2)*a^2*c^3))
/(a^2*c^3)))/(a^3*c^4) - 1/16*(6*(x*e + d)^(7/2)*c^3*d^3*e - 18*(x*e + d)^(5/2)*c^3*d^4*e + 18*(x*e + d)^(3/2)
*c^3*d^5*e - 6*sqrt(x*e + d)*c^3*d^6*e - 8*(x*e + d)^(7/2)*a*c^2*d*e^3 + 23*(x*e + d)^(5/2)*a*c^2*d^2*e^3 - 32
*(x*e + d)^(3/2)*a*c^2*d^3*e^3 + 17*sqrt(x*e + d)*a*c^2*d^4*e^3 - 9*(x*e + d)^(5/2)*a^2*c*e^5 + 14*(x*e + d)^(
3/2)*a^2*c*d*e^5 - 16*sqrt(x*e + d)*a^2*c*d^2*e^5 + 5*sqrt(x*e + d)*a^3*e^7)/(((x*e + d)^2*c - 2*(x*e + d)*c*d
+ c*d^2 - a*e^2)^2*a^2*c^2)
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maple [B] time = 0.10, size = 986, normalized size = 3.35 \[ -\frac {5 \sqrt {e x +d}\, a \,e^{7}}{16 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} c^{2}}-\frac {17 \sqrt {e x +d}\, d^{4} e^{3}}{16 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a}+\frac {3 \sqrt {e x +d}\, c \,d^{6} e}{8 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a^{2}}+\frac {\sqrt {e x +d}\, d^{2} e^{5}}{\left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} c}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} d^{3} e^{3}}{\left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a}-\frac {19 d^{2} e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {19 d^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {9 \left (e x +d \right )^{\frac {3}{2}} c \,d^{5} e}{8 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a^{2}}+\frac {3 c \,d^{4} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{8 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}+\frac {3 c \,d^{4} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{8 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} d \,e^{5}}{8 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} c}+\frac {5 e^{5} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, c}+\frac {5 e^{5} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, c}-\frac {23 \left (e x +d \right )^{\frac {5}{2}} d^{2} e^{3}}{16 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a}+\frac {9 \left (e x +d \right )^{\frac {5}{2}} c \,d^{4} e}{8 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a^{2}}+\frac {9 \left (e x +d \right )^{\frac {5}{2}} e^{5}}{16 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} c}-\frac {d \,e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a c}+\frac {d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a c}+\frac {\left (e x +d \right )^{\frac {7}{2}} d \,e^{3}}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a}-\frac {3 \left (e x +d \right )^{\frac {7}{2}} c \,d^{3} e}{8 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a^{2}}+\frac {3 d^{3} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{16 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}-\frac {3 d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{16 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(7/2)/(-c*x^2+a)^3,x)
[Out]
1/2*e^3/(c*e^2*x^2-a*e^2)^2*d/a*(e*x+d)^(7/2)-3/8*e/(c*e^2*x^2-a*e^2)^2*d^3/a^2*(e*x+d)^(7/2)*c+9/16*e^5/(c*e^
2*x^2-a*e^2)^2/c*(e*x+d)^(5/2)-23/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*d^2+9/8*e/(c*e^2*x^2-a*e^2)^2/a^2
*c*(e*x+d)^(5/2)*d^4-7/8*e^5/(c*e^2*x^2-a*e^2)^2*d/c*(e*x+d)^(3/2)+2*e^3/(c*e^2*x^2-a*e^2)^2*d^3/a*(e*x+d)^(3/
2)-9/8*e/(c*e^2*x^2-a*e^2)^2*d^5/a^2*c*(e*x+d)^(3/2)-5/16*e^7/(c*e^2*x^2-a*e^2)^2/c^2*a*(e*x+d)^(1/2)+e^5/(c*e
^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*d^2-17/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*d^4+3/8*e/(c*e^2*x^2-a*e^2)^
2*c/a^2*(e*x+d)^(1/2)*d^6+5/32*e^5/c/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c
*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)-19/32*e^3/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1
/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^2+3/8*e/a^2*c/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh
((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^4-1/4*e^3/a/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x
+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d+3/16*e/a^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2
)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^3+5/32*e^5/c/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e
*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)-19/32*e^3/a/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*a
rctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^2+3/8*e/a^2*c/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2)
)*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^4+1/4*e^3/a/c/((-c*d+(a*c*e^2)^(1/2))*c)
^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d-3/16*e/a^2/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*
arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^3
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} - a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="maxima")
[Out]
-integrate((e*x + d)^(7/2)/(c*x^2 - a)^3, x)
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mupad [B] time = 0.74, size = 2518, normalized size = 8.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^(7/2)/(a - c*x^2)^3,x)
[Out]
- ((e*(3*c*d^3 - 4*a*d*e^2)*(d + e*x)^(7/2))/(8*a^2) + ((d + e*x)^(3/2)*(7*a^2*d*e^5 + 9*c^2*d^5*e - 16*a*c*d^
3*e^3))/(8*a^2*c) + ((d + e*x)^(1/2)*(5*a^3*e^7 - 6*c^3*d^6*e + 17*a*c^2*d^4*e^3 - 16*a^2*c*d^2*e^5))/(16*a^2*
c^2) - (e*(d + e*x)^(5/2)*(9*a^2*e^4 + 18*c^2*d^4 - 23*a*c*d^2*e^2))/(16*a^2*c))/(c^2*(d + e*x)^4 + a^2*e^4 +
c^2*d^4 + (6*c^2*d^2 - 2*a*c*e^2)*(d + e*x)^2 - (4*c^2*d^3 - 4*a*c*d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 - 2*
a*c*d^2*e^2) - 2*atanh((25*e^10*(d + e*x)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e
^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) + (21*d^2*e^5*(a^
15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((825*d^5*e^9)/(2048*a^3) + (325*d*e^13)/(2048*a*c^2) - (63*c*d^7*e
^7)/(512*a^4) - (449*d^3*e^11)/(1024*a^2*c) + (125*e^14*(a^15*c^9)^(1/2))/(2048*a^8*c^7) - (95*d^2*e^12*(a^15*
c^9)^(1/2))/(512*a^9*c^6) + (381*d^4*e^10*(a^15*c^9)^(1/2))/(2048*a^10*c^5) - (63*d^6*e^8*(a^15*c^9)^(1/2))/(1
024*a^11*c^4))) - (21*d^2*e^8*(d + e*x)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4
)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) + (21*d^2*e^5*(a^15
*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((325*d*e^13)/(2048*c^3) - (63*d^7*e^7)/(512*a^3) - (449*d^3*e^11)/(1
024*a*c^2) + (825*d^5*e^9)/(2048*a^2*c) + (125*e^14*(a^15*c^9)^(1/2))/(2048*a^7*c^8) - (95*d^2*e^12*(a^15*c^9)
^(1/2))/(512*a^8*c^7) + (381*d^4*e^10*(a^15*c^9)^(1/2))/(2048*a^9*c^6) - (63*d^6*e^8*(a^15*c^9)^(1/2))/(1024*a
^10*c^5))) + (25*d*e^9*(a^15*c^9)^(1/2)*(d + e*x)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (3
85*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) + (21*d^2
*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((125*e^14*(a^15*c^9)^(1/2))/(2048*c^3) + (325*a^7*c^2*d*e^
13)/2048 - (63*a^4*c^5*d^7*e^7)/512 + (825*a^5*c^4*d^5*e^9)/2048 - (449*a^6*c^3*d^3*e^11)/1024 - (63*d^6*e^8*(
a^15*c^9)^(1/2))/(1024*a^3) - (95*d^2*e^12*(a^15*c^9)^(1/2))/(512*a*c^2) + (381*d^4*e^10*(a^15*c^9)^(1/2))/(20
48*a^2*c))) + (21*d^3*e^7*(a^15*c^9)^(1/2)*(d + e*x)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) +
(385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) + (21*
d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((63*a^5*c^4*d^7*e^7)/512 - (825*a^6*c^3*d^5*e^9)/2048 +
(449*a^7*c^2*d^3*e^11)/1024 - (125*a*e^14*(a^15*c^9)^(1/2))/(2048*c^4) - (325*a^8*c*d*e^13)/2048 + (95*d^2*e^
12*(a^15*c^9)^(1/2))/(512*c^3) - (381*d^4*e^10*(a^15*c^9)^(1/2))/(2048*a*c^2) + (63*d^6*e^8*(a^15*c^9)^(1/2))/
(1024*a^2*c))))*((144*a^5*c^8*d^7 - 25*a*e^7*(a^15*c^9)^(1/2) - 105*a^8*c^5*d*e^6 - 420*a^6*c^7*d^5*e^2 + 385*
a^7*c^6*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^9))^(1/2) - 2*atanh((25*e^10*(d + e*x)^(1/2)*((9
*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) +
(25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((825*d^
5*e^9)/(2048*a^3) + (325*d*e^13)/(2048*a*c^2) - (63*c*d^7*e^7)/(512*a^4) - (449*d^3*e^11)/(1024*a^2*c) - (125*
e^14*(a^15*c^9)^(1/2))/(2048*a^8*c^7) + (95*d^2*e^12*(a^15*c^9)^(1/2))/(512*a^9*c^6) - (381*d^4*e^10*(a^15*c^9
)^(1/2))/(2048*a^10*c^5) + (63*d^6*e^8*(a^15*c^9)^(1/2))/(1024*a^11*c^4))) - (21*d^2*e^8*(d + e*x)^(1/2)*((9*d
^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) + (
25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((325*d*e^
13)/(2048*c^3) - (63*d^7*e^7)/(512*a^3) - (449*d^3*e^11)/(1024*a*c^2) + (825*d^5*e^9)/(2048*a^2*c) - (125*e^14
*(a^15*c^9)^(1/2))/(2048*a^7*c^8) + (95*d^2*e^12*(a^15*c^9)^(1/2))/(512*a^8*c^7) - (381*d^4*e^10*(a^15*c^9)^(1
/2))/(2048*a^9*c^6) + (63*d^6*e^8*(a^15*c^9)^(1/2))/(1024*a^10*c^5))) + (25*d*e^9*(a^15*c^9)^(1/2)*(d + e*x)^(
1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^
4*c^2) + (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*
((125*e^14*(a^15*c^9)^(1/2))/(2048*c^3) - (325*a^7*c^2*d*e^13)/2048 + (63*a^4*c^5*d^7*e^7)/512 - (825*a^5*c^4*
d^5*e^9)/2048 + (449*a^6*c^3*d^3*e^11)/1024 - (63*d^6*e^8*(a^15*c^9)^(1/2))/(1024*a^3) - (95*d^2*e^12*(a^15*c^
9)^(1/2))/(512*a*c^2) + (381*d^4*e^10*(a^15*c^9)^(1/2))/(2048*a^2*c))) - (21*d^3*e^7*(a^15*c^9)^(1/2)*(d + e*x
)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024
*a^4*c^2) + (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(
32*((63*a^5*c^4*d^7*e^7)/512 - (825*a^6*c^3*d^5*e^9)/2048 + (449*a^7*c^2*d^3*e^11)/1024 + (125*a*e^14*(a^15*c^
9)^(1/2))/(2048*c^4) - (325*a^8*c*d*e^13)/2048 - (95*d^2*e^12*(a^15*c^9)^(1/2))/(512*c^3) + (381*d^4*e^10*(a^1
5*c^9)^(1/2))/(2048*a*c^2) - (63*d^6*e^8*(a^15*c^9)^(1/2))/(1024*a^2*c))))*((144*a^5*c^8*d^7 + 25*a*e^7*(a^15*
c^9)^(1/2) - 105*a^8*c^5*d*e^6 - 420*a^6*c^7*d^5*e^2 + 385*a^7*c^6*d^3*e^4 - 21*c*d^2*e^5*(a^15*c^9)^(1/2))/(4
096*a^10*c^9))^(1/2)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**(7/2)/(-c*x**2+a)**3,x)
[Out]
Timed out
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