3.2.4 \(\int \frac {(c+d x^2)^{3/2}}{(a+b x^2)^4} \, dx\) [104]

Optimal. Leaf size=199 \[ \frac {c (5 b c-6 a d) x \sqrt {c+d x^2}}{16 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {(5 b c-6 a d) x \left (c+d x^2\right )^{3/2}}{24 a^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a (b c-a d) \left (a+b x^2\right )^3}+\frac {c^2 (5 b c-6 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{16 a^{7/2} (b c-a d)^{3/2}} \]

[Out]

1/24*(-6*a*d+5*b*c)*x*(d*x^2+c)^(3/2)/a^2/(-a*d+b*c)/(b*x^2+a)^2+1/6*b*x*(d*x^2+c)^(5/2)/a/(-a*d+b*c)/(b*x^2+a
)^3+1/16*c^2*(-6*a*d+5*b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^(7/2)/(-a*d+b*c)^(3/2)+1/16*c
*(-6*a*d+5*b*c)*x*(d*x^2+c)^(1/2)/a^3/(-a*d+b*c)/(b*x^2+a)

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Rubi [A]
time = 0.07, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {390, 386, 385, 211} \begin {gather*} \frac {c^2 (5 b c-6 a d) \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{16 a^{7/2} (b c-a d)^{3/2}}+\frac {c x \sqrt {c+d x^2} (5 b c-6 a d)}{16 a^3 \left (a+b x^2\right ) (b c-a d)}+\frac {x \left (c+d x^2\right )^{3/2} (5 b c-6 a d)}{24 a^2 \left (a+b x^2\right )^2 (b c-a d)}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a \left (a+b x^2\right )^3 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^(3/2)/(a + b*x^2)^4,x]

[Out]

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

Rule 211

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 386

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[(-x)*(a + b*x^n)^(p + 1)*(
(c + d*x^n)^q/(a*n*(p + 1))), x] - Dist[c*(q/(a*(p + 1))), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 15.22, size = 179, normalized size = 0.90 \begin {gather*} \frac {-\frac {\sqrt {a} x \sqrt {c+d x^2} \left (15 b^3 c^2 x^4+8 a b^2 c x^2 \left (5 c-d x^2\right )-6 a^3 d \left (5 c+2 d x^2\right )+a^2 b \left (33 c^2-22 c d x^2-4 d^2 x^4\right )\right )}{(-b c+a d) \left (a+b x^2\right )^3}+\frac {3 c^2 (5 b c-6 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{3/2}}}{48 a^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^(3/2)/(a + b*x^2)^4,x]

[Out]

(-((Sqrt[a]*x*Sqrt[c + d*x^2]*(15*b^3*c^2*x^4 + 8*a*b^2*c*x^2*(5*c - d*x^2) - 6*a^3*d*(5*c + 2*d*x^2) + a^2*b*
(33*c^2 - 22*c*d*x^2 - 4*d^2*x^4)))/((-(b*c) + a*d)*(a + b*x^2)^3)) + (3*c^2*(5*b*c - 6*a*d)*ArcTan[(Sqrt[b*c
- a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*c - a*d)^(3/2))/(48*a^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(12957\) vs. \(2(175)=350\).
time = 0.09, size = 12958, normalized size = 65.12

method	result	size
default	\(\text {Expression too large to display}\)	\(12958\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^(3/2)/(b*x^2+a)^4,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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((d*x^2+c)^(3/2)/(b*x^2+a)^4,x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)^(3/2)/(b*x^2 + a)^4, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (175) = 350\).
time = 1.98, size = 972, normalized size = 4.88 \begin {gather*} \left [-\frac {3 \, {\left (5 \, a^{3} b c^{3} - 6 \, a^{4} c^{2} d + {\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{6} + 3 \, {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{4} + 3 \, {\left (5 \, a^{2} b^{2} c^{3} - 6 \, a^{3} b c^{2} d\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{5} + 2 \, {\left (20 \, a^{2} b^{3} c^{3} - 31 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} + 6 \, a^{5} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a^{3} b^{2} c^{3} - 21 \, a^{4} b c^{2} d + 10 \, a^{5} c d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{192 \, {\left (a^{7} b^{2} c^{2} - 2 \, a^{8} b c d + a^{9} d^{2} + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} x^{6} + 3 \, {\left (a^{5} b^{4} c^{2} - 2 \, a^{6} b^{3} c d + a^{7} b^{2} d^{2}\right )} x^{4} + 3 \, {\left (a^{6} b^{3} c^{2} - 2 \, a^{7} b^{2} c d + a^{8} b d^{2}\right )} x^{2}\right )}}, \frac {3 \, {\left (5 \, a^{3} b c^{3} - 6 \, a^{4} c^{2} d + {\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{6} + 3 \, {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{4} + 3 \, {\left (5 \, a^{2} b^{2} c^{3} - 6 \, a^{3} b c^{2} d\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{5} + 2 \, {\left (20 \, a^{2} b^{3} c^{3} - 31 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} + 6 \, a^{5} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a^{3} b^{2} c^{3} - 21 \, a^{4} b c^{2} d + 10 \, a^{5} c d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{96 \, {\left (a^{7} b^{2} c^{2} - 2 \, a^{8} b c d + a^{9} d^{2} + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} x^{6} + 3 \, {\left (a^{5} b^{4} c^{2} - 2 \, a^{6} b^{3} c d + a^{7} b^{2} d^{2}\right )} x^{4} + 3 \, {\left (a^{6} b^{3} c^{2} - 2 \, a^{7} b^{2} c d + a^{8} b d^{2}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(3/2)/(b*x^2+a)^4,x, algorithm="fricas")

[Out]

[-1/192*(3*(5*a^3*b*c^3 - 6*a^4*c^2*d + (5*b^4*c^3 - 6*a*b^3*c^2*d)*x^6 + 3*(5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*x^
4 + 3*(5*a^2*b^2*c^3 - 6*a^3*b*c^2*d)*x^2)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a
^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(
b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*((15*a*b^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^5 + 2*(20
*a^2*b^3*c^3 - 31*a^3*b^2*c^2*d + 5*a^4*b*c*d^2 + 6*a^5*d^3)*x^3 + 3*(11*a^3*b^2*c^3 - 21*a^4*b*c^2*d + 10*a^5
*c*d^2)*x)*sqrt(d*x^2 + c))/(a^7*b^2*c^2 - 2*a^8*b*c*d + a^9*d^2 + (a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)
*x^6 + 3*(a^5*b^4*c^2 - 2*a^6*b^3*c*d + a^7*b^2*d^2)*x^4 + 3*(a^6*b^3*c^2 - 2*a^7*b^2*c*d + a^8*b*d^2)*x^2), 1
/96*(3*(5*a^3*b*c^3 - 6*a^4*c^2*d + (5*b^4*c^3 - 6*a*b^3*c^2*d)*x^6 + 3*(5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*x^4 +
3*(5*a^2*b^2*c^3 - 6*a^3*b*c^2*d)*x^2)*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 -
a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + 2*((15*a*b^4*c^3 - 23*a^2*b^3*c^2*d
+ 4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^5 + 2*(20*a^2*b^3*c^3 - 31*a^3*b^2*c^2*d + 5*a^4*b*c*d^2 + 6*a^5*d^3)*x^3
+ 3*(11*a^3*b^2*c^3 - 21*a^4*b*c^2*d + 10*a^5*c*d^2)*x)*sqrt(d*x^2 + c))/(a^7*b^2*c^2 - 2*a^8*b*c*d + a^9*d^2
+ (a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*x^6 + 3*(a^5*b^4*c^2 - 2*a^6*b^3*c*d + a^7*b^2*d^2)*x^4 + 3*(a^6
*b^3*c^2 - 2*a^7*b^2*c*d + a^8*b*d^2)*x^2)]

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Sympy [F(-1)] Timed out
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((d*x**2+c)**(3/2)/(b*x**2+a)**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 919 vs. \(2 (175) = 350\).
time = 1.62, size = 919, normalized size = 4.62 \begin {gather*} -\frac {{\left (5 \, b c^{3} \sqrt {d} - 6 \, a c^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{16 \, {\left (a^{3} b c - a^{4} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{5} c^{3} \sqrt {d} - 18 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b^{4} c^{2} d^{\frac {3}{2}} - 75 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{5} c^{4} \sqrt {d} + 240 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b^{4} c^{3} d^{\frac {3}{2}} - 180 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} b^{3} c^{2} d^{\frac {5}{2}} - 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{3} b^{2} c d^{\frac {7}{2}} + 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{4} b d^{\frac {9}{2}} + 150 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{5} c^{5} \sqrt {d} - 620 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b^{4} c^{4} d^{\frac {3}{2}} + 968 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} b^{3} c^{3} d^{\frac {5}{2}} - 720 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{3} b^{2} c^{2} d^{\frac {7}{2}} + 64 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{4} b c d^{\frac {9}{2}} + 128 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{5} d^{\frac {11}{2}} - 150 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{5} c^{6} \sqrt {d} + 600 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b^{4} c^{5} d^{\frac {3}{2}} - 864 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} b^{3} c^{4} d^{\frac {5}{2}} + 288 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{3} b^{2} c^{3} d^{\frac {7}{2}} + 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{4} b c^{2} d^{\frac {9}{2}} + 75 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{5} c^{7} \sqrt {d} - 210 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{4} c^{6} d^{\frac {3}{2}} + 72 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b^{3} c^{5} d^{\frac {5}{2}} + 48 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} b^{2} c^{4} d^{\frac {7}{2}} - 15 \, b^{5} c^{8} \sqrt {d} + 8 \, a b^{4} c^{7} d^{\frac {3}{2}} + 4 \, a^{2} b^{3} c^{6} d^{\frac {5}{2}}}{24 \, {\left (a^{3} b^{3} c - a^{4} b^{2} d\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(3/2)/(b*x^2+a)^4,x, algorithm="giac")

[Out]

-1/16*(5*b*c^3*sqrt(d) - 6*a*c^2*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*
b*c*d - a^2*d^2))/((a^3*b*c - a^4*d)*sqrt(a*b*c*d - a^2*d^2)) - 1/24*(15*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^5*
c^3*sqrt(d) - 18*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b^4*c^2*d^(3/2) - 75*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^5*c
^4*sqrt(d) + 240*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b^4*c^3*d^(3/2) - 180*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*b
^3*c^2*d^(5/2) - 96*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^3*b^2*c*d^(7/2) + 96*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^4
*b*d^(9/2) + 150*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^5*c^5*sqrt(d) - 620*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b^4*c
^4*d^(3/2) + 968*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*b^3*c^3*d^(5/2) - 720*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^3
*b^2*c^2*d^(7/2) + 64*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^4*b*c*d^(9/2) + 128*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^
5*d^(11/2) - 150*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^5*c^6*sqrt(d) + 600*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b^4*c
^5*d^(3/2) - 864*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*b^3*c^4*d^(5/2) + 288*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^3
*b^2*c^3*d^(7/2) + 96*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^4*b*c^2*d^(9/2) + 75*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b
^5*c^7*sqrt(d) - 210*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b^4*c^6*d^(3/2) + 72*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^
2*b^3*c^5*d^(5/2) + 48*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^3*b^2*c^4*d^(7/2) - 15*b^5*c^8*sqrt(d) + 8*a*b^4*c^7*
d^(3/2) + 4*a^2*b^3*c^6*d^(5/2))/((a^3*b^3*c - a^4*b^2*d)*((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x -
sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d\,x^2+c\right )}^{3/2}}{{\left (b\,x^2+a\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^(3/2)/(a + b*x^2)^4,x)

[Out]

int((c + d*x^2)^(3/2)/(a + b*x^2)^4, x)

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