3.6.55 \(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^7} \, dx\) [555]

Optimal. Leaf size=203 \[ -\frac {5 a^2 c^2 (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {5 a^3 c^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{7/2}} \]

[Out]

-5/24*a*c*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^2/(e*x+d)^4-1/6*(-c*d*x+a*e)*(c*x^2+a)^(5/2)/(a*e^2+c*d^2
)/(e*x+d)^6-5/16*a^3*c^3*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/(a*e^2+c*d^2)^(7/2)-5/16*a^
2*c^2*(-c*d*x+a*e)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/(e*x+d)^2

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Rubi [A]
time = 0.07, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {735, 739, 212} \begin {gather*} -\frac {5 a^3 c^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{7/2}}-\frac {5 a^2 c^2 \sqrt {a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {5 a c \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac {\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

(-5*a^2*c^2*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (5*a*c*(a*e - c*d*x)*(a + c*x^
2)^(3/2))/(24*(c*d^2 + a*e^2)^2*(d + e*x)^4) - ((a*e - c*d*x)*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)*(d + e*x)^
6) - (5*a^3*c^3*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 735

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin {align*} \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx &=-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}+\frac {(5 a c) \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{6 \left (c d^2+a e^2\right )}\\ &=-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}+\frac {\left (5 a^2 c^2\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{8 \left (c d^2+a e^2\right )^2}\\ &=-\frac {5 a^2 c^2 (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}+\frac {\left (5 a^3 c^3\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{16 \left (c d^2+a e^2\right )^3}\\ &=-\frac {5 a^2 c^2 (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {\left (5 a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^3}\\ &=-\frac {5 a^2 c^2 (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {5 a^3 c^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 10.40, size = 305, normalized size = 1.50 \begin {gather*} \frac {1}{48} \left (\frac {\sqrt {a+c x^2} \left (-8 a^5 e^5+8 c^5 d^5 x^5-2 a^4 c e^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )+2 a c^4 d^3 x^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )-a^3 c^2 e \left (33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right )+a^2 c^3 d x \left (33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right )\right )}{\left (c d^2+a e^2\right )^3 (d+e x)^6}+\frac {15 a^3 c^3 \log (d+e x)}{\left (c d^2+a e^2\right )^{7/2}}-\frac {15 a^3 c^3 \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{7/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

((Sqrt[a + c*x^2]*(-8*a^5*e^5 + 8*c^5*d^5*x^5 - 2*a^4*c*e^3*(13*d^2 + 6*d*e*x + 13*e^2*x^2) + 2*a*c^4*d^3*x^3*
(13*d^2 + 6*d*e*x + 13*e^2*x^2) - a^3*c^2*e*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4
) + a^2*c^3*d*x*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4)))/((c*d^2 + a*e^2)^3*(d +
e*x)^6) + (15*a^3*c^3*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2) - (15*a^3*c^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*
Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))/48

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(15867\) vs. \(2(183)=366\).
time = 0.53, size = 15868, normalized size = 78.17

method	result	size
default	\(\text {Expression too large to display}\)	\(15868\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 4385 vs. \(2 (186) = 372\).
time = 0.57, size = 4385, normalized size = 21.60 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="maxima")

[Out]

-5/32*c^8*d^9*arcsinh(c*x/sqrt(a*c))/(c^(11/2)*d^10*e^6 + 5*a*c^(9/2)*d^8*e^8 + 10*a^2*c^(7/2)*d^6*e^10 + 10*a
^3*c^(5/2)*d^4*e^12 + 5*a^4*c^(3/2)*d^2*e^14 + a^5*sqrt(c)*e^16) - 5/32*a*c^7*d^7*arcsinh(c*x/sqrt(a*c))/(c^(1
1/2)*d^10*e^4 + 5*a*c^(9/2)*d^8*e^6 + 10*a^2*c^(7/2)*d^6*e^8 + 10*a^3*c^(5/2)*d^4*e^10 + 5*a^4*c^(3/2)*d^2*e^1
2 + a^5*sqrt(c)*e^14) + 5/32*sqrt(c*x^2 + a)*c^7*d^7*x/(c^5*d^10*e^4 + 5*a*c^4*d^8*e^6 + 10*a^2*c^3*d^6*e^8 +
10*a^3*c^2*d^4*e^10 + 5*a^4*c*d^2*e^12 + a^5*e^14) + 15/16*c^7*d^7*arcsinh(c*x/sqrt(a*c))/(c^(9/2)*d^8*e^6 + 4
*a*c^(7/2)*d^6*e^8 + 6*a^2*c^(5/2)*d^4*e^10 + 4*a^3*c^(3/2)*d^2*e^12 + a^4*sqrt(c)*e^14) - 5/48*(c*x^2 + a)^(3
/2)*c^6*d^6/(c^5*d^10*e^3 + 5*a*c^4*d^8*e^5 + 10*a^2*c^3*d^6*e^7 + 10*a^3*c^2*d^4*e^9 + 5*a^4*c*d^2*e^11 + a^5
*e^13) + 5/48*(c*x^2 + a)^(3/2)*c^6*d^5*x/(c^5*d^10*e^2 + 5*a*c^4*d^8*e^4 + 10*a^2*c^3*d^6*e^6 + 10*a^3*c^2*d^
4*e^8 + 5*a^4*c*d^2*e^10 + a^5*e^12) + 5/32*sqrt(c*x^2 + a)*a*c^6*d^5*x/(c^5*d^10*e^2 + 5*a*c^4*d^8*e^4 + 10*a
^2*c^3*d^6*e^6 + 10*a^3*c^2*d^4*e^8 + 5*a^4*c*d^2*e^10 + a^5*e^12) + 25/32*a*c^6*d^5*arcsinh(c*x/sqrt(a*c))/(c
^(9/2)*d^8*e^4 + 4*a*c^(7/2)*d^6*e^6 + 6*a^2*c^(5/2)*d^4*e^8 + 4*a^3*c^(3/2)*d^2*e^10 + a^4*sqrt(c)*e^12) - 1/
16*(c*x^2 + a)^(5/2)*c^5*d^5/(c^5*d^10*x*e^2 + c^5*d^11*e + 5*a*c^4*d^8*x*e^4 + 5*a*c^4*d^9*e^3 + 10*a^2*c^3*d
^6*x*e^6 + 10*a^2*c^3*d^7*e^5 + 10*a^3*c^2*d^4*x*e^8 + 10*a^3*c^2*d^5*e^7 + 5*a^4*c*d^2*x*e^10 + 5*a^4*c*d^3*e
^9 + a^5*x*e^12 + a^5*d*e^11) - 5/16*sqrt(c*x^2 + a)*c^6*d^6/(c^4*d^8*e^5 + 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d^4*e^
9 + 4*a^3*c*d^2*e^11 + a^4*e^13) - 15/32*sqrt(c*x^2 + a)*c^6*d^5*x/(c^4*d^8*e^4 + 4*a*c^3*d^6*e^6 + 6*a^2*c^2*
d^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12) - 15/8*c^6*d^5*arcsinh(c*x/sqrt(a*c))/(c^(7/2)*d^6*e^6 + 3*a*c^(5/2)*d^
4*e^8 + 3*a^2*c^(3/2)*d^2*e^10 + a^3*sqrt(c)*e^12) + 1/48*(c*x^2 + a)^(7/2)*c^4*d^4/(c^5*d^10*x^2*e + c^5*d^12
*e^(-1) + 2*c^5*d^11*x + 5*a*c^4*d^8*x^2*e^3 + 10*a*c^4*d^9*x*e^2 + 5*a*c^4*d^10*e + 10*a^2*c^3*d^6*x^2*e^5 +
20*a^2*c^3*d^7*x*e^4 + 10*a^2*c^3*d^8*e^3 + 10*a^3*c^2*d^4*x^2*e^7 + 20*a^3*c^2*d^5*x*e^6 + 10*a^3*c^2*d^6*e^5
+ 5*a^4*c*d^2*x^2*e^9 + 10*a^4*c*d^3*x*e^8 + 5*a^4*c*d^4*e^7 + a^5*x^2*e^11 + 2*a^5*d*x*e^10 + a^5*d^2*e^9) -
1/48*(c*x^2 + a)^(5/2)*c^5*d^4/(c^5*d^10*e + 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 + 10*a^3*c^2*d^4*e^7 + 5*a^
4*c*d^2*e^9 + a^5*e^11) - 5/16*c^6*d^6*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*
e^(-13)/(c*d^2*e^(-2) + a)^(7/2) + 5/16*(c*x^2 + a)^(3/2)*c^5*d^4/(c^4*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d
^4*e^7 + 4*a^3*c*d^2*e^9 + a^4*e^11) - 5/12*(c*x^2 + a)^(3/2)*c^5*d^3*x/(c^4*d^8*e^2 + 4*a*c^3*d^6*e^4 + 6*a^2
*c^2*d^4*e^6 + 4*a^3*c*d^2*e^8 + a^4*e^10) - 5/8*sqrt(c*x^2 + a)*a*c^5*d^3*x/(c^4*d^8*e^2 + 4*a*c^3*d^6*e^4 +
6*a^2*c^2*d^4*e^6 + 4*a^3*c*d^2*e^8 + a^4*e^10) - 35/32*a*c^5*d^3*arcsinh(c*x/sqrt(a*c))/(c^(7/2)*d^6*e^4 + 3*
a*c^(5/2)*d^4*e^6 + 3*a^2*c^(3/2)*d^2*e^8 + a^3*sqrt(c)*e^10) - 1/24*(c*x^2 + a)^(7/2)*c^3*d^3/(c^4*d^8*x^3*e^
2 + 3*c^4*d^9*x^2*e + c^4*d^11*e^(-1) + 3*c^4*d^10*x + 4*a*c^3*d^6*x^3*e^4 + 12*a*c^3*d^7*x^2*e^3 + 12*a*c^3*d
^8*x*e^2 + 4*a*c^3*d^9*e + 6*a^2*c^2*d^4*x^3*e^6 + 18*a^2*c^2*d^5*x^2*e^5 + 18*a^2*c^2*d^6*x*e^4 + 6*a^2*c^2*d
^7*e^3 + 4*a^3*c*d^2*x^3*e^8 + 12*a^3*c*d^3*x^2*e^7 + 12*a^3*c*d^4*x*e^6 + 4*a^3*c*d^5*e^5 + a^4*x^3*e^10 + 3*
a^4*d*x^2*e^9 + 3*a^4*d^2*x*e^8 + a^4*d^3*e^7) + 5/24*(c*x^2 + a)^(5/2)*c^4*d^3/(c^4*d^8*x*e^2 + c^4*d^9*e + 4
*a*c^3*d^6*x*e^4 + 4*a*c^3*d^7*e^3 + 6*a^2*c^2*d^4*x*e^6 + 6*a^2*c^2*d^5*e^5 + 4*a^3*c*d^2*x*e^8 + 4*a^3*c*d^3
*e^7 + a^4*x*e^10 + a^4*d*e^9) + 15/16*sqrt(c*x^2 + a)*c^5*d^4/(c^3*d^6*e^5 + 3*a*c^2*d^4*e^7 + 3*a^2*c*d^2*e^
9 + a^3*e^11) + 15/32*sqrt(c*x^2 + a)*c^5*d^3*x/(c^3*d^6*e^4 + 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 + a^3*e^10) +
15/16*c^5*d^4*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-11)/(c*d^2*e^(-2) +
a)^(5/2) + 25/16*c^5*d^3*arcsinh(c*x/sqrt(a*c))/(c^(5/2)*d^4*e^6 + 2*a*c^(3/2)*d^2*e^8 + a^2*sqrt(c)*e^10) - 1
/8*(c*x^2 + a)^(7/2)*c^3*d^2/(c^4*d^8*x^2*e + c^4*d^10*e^(-1) + 2*c^4*d^9*x + 4*a*c^3*d^6*x^2*e^3 + 8*a*c^3*d^
7*x*e^2 + 4*a*c^3*d^8*e + 6*a^2*c^2*d^4*x^2*e^5 + 12*a^2*c^2*d^5*x*e^4 + 6*a^2*c^2*d^6*e^3 + 4*a^3*c*d^2*x^2*e
^7 + 8*a^3*c*d^3*x*e^6 + 4*a^3*c*d^4*e^5 + a^4*x^2*e^9 + 2*a^4*d*x*e^8 + a^4*d^2*e^7) + 1/8*(c*x^2 + a)^(5/2)*
c^4*d^2/(c^4*d^8*e + 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9) - 1/8*(c*x^2 + a)^(7/2)*
c^2*d^2/(c^3*d^6*x^4*e^3 + 4*c^3*d^7*x^3*e^2 + 6*c^3*d^8*x^2*e + c^3*d^10*e^(-1) + 4*c^3*d^9*x + 3*a*c^2*d^4*x
^4*e^5 + 12*a*c^2*d^5*x^3*e^4 + 18*a*c^2*d^6*x^2*e^3 + 12*a*c^2*d^7*x*e^2 + 3*a*c^2*d^8*e + 3*a^2*c*d^2*x^4*e^
7 + 12*a^2*c*d^3*x^3*e^6 + 18*a^2*c*d^4*x^2*e^5 + 12*a^2*c*d^5*x*e^4 + 3*a^2*c*d^6*e^3 + a^3*x^4*e^9 + 4*a^3*d
*x^3*e^8 + 6*a^3*d^2*x^2*e^7 + 4*a^3*d^3*x*e^6 + a^3*d^4*e^5) - 5/16*(c*x^2 + a)^(3/2)*c^4*d^2/(c^3*d^6*e^3 +
3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 + a^3*e^9) + 5/16*(c*x^2 + a)^(3/2)*c^4*d*x/(c^3*d^6*e^2 + 3*a*c^2*d^4*e^4 +
3*a^2*c*d^2*e^6 + a^3*e^8) + 15/32*sqrt(c*x^2 ...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 955 vs. \(2 (186) = 372\).
time = 15.70, size = 1937, normalized size = 9.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

[1/96*(15*(a^3*c^3*x^6*e^6 + 6*a^3*c^3*d*x^5*e^5 + 15*a^3*c^3*d^2*x^4*e^4 + 20*a^3*c^3*d^3*x^3*e^3 + 15*a^3*c^
3*d^4*x^2*e^2 + 6*a^3*c^3*d^5*x*e + a^3*c^3*d^6)*sqrt(c*d^2 + a*e^2)*log(-(2*c^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d
^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*x^2 + 2*a^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) +
2*(8*c^6*d^7*x^5 + 26*a*c^5*d^7*x^3 + 33*a^2*c^4*d^7*x - (33*a^4*c^2*x^4 + 26*a^5*c*x^2 + 8*a^6)*e^7 + 3*(11*
a^3*c^3*d*x^5 - 18*a^4*c^2*d*x^3 - 4*a^5*c*d*x)*e^6 + (21*a^3*c^3*d^2*x^4 - 148*a^4*c^2*d^2*x^2 - 34*a^5*c*d^2
)*e^5 + (59*a^2*c^4*d^3*x^5 + 68*a^3*c^3*d^3*x^3 - 66*a^4*c^2*d^3*x)*e^4 + (66*a^2*c^4*d^4*x^4 - 68*a^3*c^3*d^
4*x^2 - 59*a^4*c^2*d^4)*e^3 + (34*a*c^5*d^5*x^5 + 148*a^2*c^4*d^5*x^3 - 21*a^3*c^3*d^5*x)*e^2 + 3*(4*a*c^5*d^6
*x^4 + 18*a^2*c^4*d^6*x^2 - 11*a^3*c^3*d^6)*e)*sqrt(c*x^2 + a))/(6*c^4*d^13*x*e + c^4*d^14 + a^4*x^6*e^14 + 6*
a^4*d*x^5*e^13 + (4*a^3*c*d^2*x^6 + 15*a^4*d^2*x^4)*e^12 + 4*(6*a^3*c*d^3*x^5 + 5*a^4*d^3*x^3)*e^11 + 3*(2*a^2
*c^2*d^4*x^6 + 20*a^3*c*d^4*x^4 + 5*a^4*d^4*x^2)*e^10 + 2*(18*a^2*c^2*d^5*x^5 + 40*a^3*c*d^5*x^3 + 3*a^4*d^5*x
)*e^9 + (4*a*c^3*d^6*x^6 + 90*a^2*c^2*d^6*x^4 + 60*a^3*c*d^6*x^2 + a^4*d^6)*e^8 + 24*(a*c^3*d^7*x^5 + 5*a^2*c^
2*d^7*x^3 + a^3*c*d^7*x)*e^7 + (c^4*d^8*x^6 + 60*a*c^3*d^8*x^4 + 90*a^2*c^2*d^8*x^2 + 4*a^3*c*d^8)*e^6 + 2*(3*
c^4*d^9*x^5 + 40*a*c^3*d^9*x^3 + 18*a^2*c^2*d^9*x)*e^5 + 3*(5*c^4*d^10*x^4 + 20*a*c^3*d^10*x^2 + 2*a^2*c^2*d^1
0)*e^4 + 4*(5*c^4*d^11*x^3 + 6*a*c^3*d^11*x)*e^3 + (15*c^4*d^12*x^2 + 4*a*c^3*d^12)*e^2), 1/48*(15*(a^3*c^3*x^
6*e^6 + 6*a^3*c^3*d*x^5*e^5 + 15*a^3*c^3*d^2*x^4*e^4 + 20*a^3*c^3*d^3*x^3*e^3 + 15*a^3*c^3*d^4*x^2*e^2 + 6*a^3
*c^3*d^5*x*e + a^3*c^3*d^6)*sqrt(-c*d^2 - a*e^2)*arctan(-sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(c
^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + a^2)*e^2)) + (8*c^6*d^7*x^5 + 26*a*c^5*d^7*x^3 + 33*a^2*c^4*d^7*x - (33*a^4*
c^2*x^4 + 26*a^5*c*x^2 + 8*a^6)*e^7 + 3*(11*a^3*c^3*d*x^5 - 18*a^4*c^2*d*x^3 - 4*a^5*c*d*x)*e^6 + (21*a^3*c^3*
d^2*x^4 - 148*a^4*c^2*d^2*x^2 - 34*a^5*c*d^2)*e^5 + (59*a^2*c^4*d^3*x^5 + 68*a^3*c^3*d^3*x^3 - 66*a^4*c^2*d^3*
x)*e^4 + (66*a^2*c^4*d^4*x^4 - 68*a^3*c^3*d^4*x^2 - 59*a^4*c^2*d^4)*e^3 + (34*a*c^5*d^5*x^5 + 148*a^2*c^4*d^5*
x^3 - 21*a^3*c^3*d^5*x)*e^2 + 3*(4*a*c^5*d^6*x^4 + 18*a^2*c^4*d^6*x^2 - 11*a^3*c^3*d^6)*e)*sqrt(c*x^2 + a))/(6
*c^4*d^13*x*e + c^4*d^14 + a^4*x^6*e^14 + 6*a^4*d*x^5*e^13 + (4*a^3*c*d^2*x^6 + 15*a^4*d^2*x^4)*e^12 + 4*(6*a^
3*c*d^3*x^5 + 5*a^4*d^3*x^3)*e^11 + 3*(2*a^2*c^2*d^4*x^6 + 20*a^3*c*d^4*x^4 + 5*a^4*d^4*x^2)*e^10 + 2*(18*a^2*
c^2*d^5*x^5 + 40*a^3*c*d^5*x^3 + 3*a^4*d^5*x)*e^9 + (4*a*c^3*d^6*x^6 + 90*a^2*c^2*d^6*x^4 + 60*a^3*c*d^6*x^2 +
a^4*d^6)*e^8 + 24*(a*c^3*d^7*x^5 + 5*a^2*c^2*d^7*x^3 + a^3*c*d^7*x)*e^7 + (c^4*d^8*x^6 + 60*a*c^3*d^8*x^4 + 9
0*a^2*c^2*d^8*x^2 + 4*a^3*c*d^8)*e^6 + 2*(3*c^4*d^9*x^5 + 40*a*c^3*d^9*x^3 + 18*a^2*c^2*d^9*x)*e^5 + 3*(5*c^4*
d^10*x^4 + 20*a*c^3*d^10*x^2 + 2*a^2*c^2*d^10)*e^4 + 4*(5*c^4*d^11*x^3 + 6*a*c^3*d^11*x)*e^3 + (15*c^4*d^12*x^
2 + 4*a*c^3*d^12)*e^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**7,x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**7, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1895 vs. \(2 (186) = 372\).
time = 2.38, size = 1895, normalized size = 9.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

5/8*a^3*c^3*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*d^
4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) + 1/24*(768*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*d^10*
e + 256*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(17/2)*d^11 + 960*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(15/2)*d^9*e^2 +
640*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*d^8*e^3 - 768*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^8*d^10*e + 240*(sqr
t(c)*x - sqrt(c*x^2 + a))^10*c^(13/2)*d^7*e^4 - 1088*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(15/2)*d^9*e^2 + 48*(
sqrt(c)*x - sqrt(c*x^2 + a))^11*c^6*d^6*e^5 + 576*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^7*d^8*e^3 + 2160*(sqrt(c
)*x - sqrt(c*x^2 + a))^8*a*c^(13/2)*d^7*e^4 + 960*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(15/2)*d^9*e^2 + 1840*
(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*d^6*e^5 - 576*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^7*d^8*e^3 + 720*(sqr
t(c)*x - sqrt(c*x^2 + a))^10*a*c^(11/2)*d^5*e^6 - 3744*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*d^7*e^4 +
144*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^5*d^4*e^7 - 2592*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*d^6*e^5 - 64
0*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^7*d^8*e^3 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*d^5*e^6 +
2160*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(13/2)*d^7*e^4 + 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^5*d^4*
e^7 + 2592*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^6*d^6*e^5 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2)*
d^3*e^8 - 3320*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*d^5*e^6 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*
c^(13/2)*d^7*e^4 + 144*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^4*d^2*e^9 - 5640*(sqrt(c)*x - sqrt(c*x^2 + a))^7
*a^3*c^5*d^4*e^7 - 1840*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^6*d^6*e^5 - 2910*(sqrt(c)*x - sqrt(c*x^2 + a))^8
*a^3*c^(9/2)*d^3*e^8 + 1080*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(11/2)*d^5*e^6 - 340*(sqrt(c)*x - sqrt(c*x^2
+ a))^9*a^3*c^4*d^2*e^9 + 7080*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^5*d^4*e^7 - 48*(sqrt(c)*x - sqrt(c*x^2 +
a))*a^5*c^6*d^6*e^5 + 75*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(7/2)*d*e^10 + 5680*(sqrt(c)*x - sqrt(c*x^2 +
a))^6*a^4*c^(9/2)*d^3*e^8 + 792*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(11/2)*d^5*e^6 + 33*(sqrt(c)*x - sqrt(c
*x^2 + a))^11*a^3*c^3*e^11 + 1800*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^4*d^2*e^9 - 2040*(sqrt(c)*x - sqrt(c*x
^2 + a))^3*a^5*c^5*d^4*e^7 + 45*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(7/2)*d*e^10 - 4620*(sqrt(c)*x - sqrt(c*
x^2 + a))^4*a^5*c^(9/2)*d^3*e^8 + 8*a^6*c^(11/2)*d^5*e^6 + 5*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^3*e^11 - 21
60*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*d^2*e^9 - 168*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^5*d^4*e^7 - 330*(
sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(7/2)*d*e^10 + 1104*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(9/2)*d^3*e^8 +
90*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^3*e^11 + 1640*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^4*d^2*e^9 + 450*
(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(7/2)*d*e^10 + 26*a^7*c^(9/2)*d^3*e^8 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))
^5*a^6*c^3*e^11 - 252*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^4*d^2*e^9 - 273*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*
c^(7/2)*d*e^10 + 5*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^3*e^11 + 33*a^8*c^(7/2)*d*e^10 + 33*(sqrt(c)*x - sqrt
(c*x^2 + a))*a^8*c^3*e^11)/((c^3*d^6*e^6 + 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 + a^3*e^12)*((sqrt(c)*x - sqrt(c
*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^6)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^(5/2)/(d + e*x)^7,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^7, x)

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