3.1.81 \(\int \frac {(c+d x^2)^4}{(a+b x^2)^{3/2}} \, dx\) [81]

Optimal. Leaf size=257 \[ -\frac {d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt {a+b x^2}}{48 a b^4}-\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}} \]

[Out]

1/16*d*(-35*a^3*d^3+120*a^2*b*c*d^2-144*a*b^2*c^2*d+64*b^3*c^3)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(9/2)+(-a
*d+b*c)*x*(d*x^2+c)^3/a/b/(b*x^2+a)^(1/2)-1/48*d*(-105*a^3*d^3+290*a^2*b*c*d^2-248*a*b^2*c^2*d+48*b^3*c^3)*x*(
b*x^2+a)^(1/2)/a/b^4-1/24*d*(35*a^2*d^2-64*a*b*c*d+24*b^2*c^2)*x*(d*x^2+c)*(b*x^2+a)^(1/2)/a/b^3-1/6*d*(-7*a*d
+6*b*c)*x*(d*x^2+c)^2*(b*x^2+a)^(1/2)/a/b^2

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Rubi [A]
time = 0.18, antiderivative size = 257, 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 = {424, 542, 396, 223, 212} \begin {gather*} -\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (35 a^2 d^2-64 a b c d+24 b^2 c^2\right )}{24 a b^3}+\frac {d \left (-35 a^3 d^3+120 a^2 b c d^2-144 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}}-\frac {d x \sqrt {a+b x^2} \left (-105 a^3 d^3+290 a^2 b c d^2-248 a b^2 c^2 d+48 b^3 c^3\right )}{48 a b^4}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2 (6 b c-7 a d)}{6 a b^2}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{a b \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/48*(d*(48*b^3*c^3 - 248*a*b^2*c^2*d + 290*a^2*b*c*d^2 - 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(a*b^4) - (d*(24*b^
2*c^2 - 64*a*b*c*d + 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(24*a*b^3) - (d*(6*b*c - 7*a*d)*x*Sqrt[a + b*x
^2]*(c + d*x^2)^2)/(6*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^3)/(a*b*Sqrt[a + b*x^2]) + (d*(64*b^3*c^3 - 144*a*b^
2*c^2*d + 120*a^2*b*c*d^2 - 35*a^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(9/2))

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 396

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

Rule 424

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

Rule 542

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

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\int \frac {\left (c+d x^2\right )^2 \left (a c d-d (6 b c-7 a d) x^2\right )}{\sqrt {a+b x^2}} \, dx}{a b}\\ &=-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\int \frac {\left (c+d x^2\right ) \left (a c d (12 b c-7 a d)-d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx}{6 a b^2}\\ &=-\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\int \frac {a c d \left (72 b^2 c^2-92 a b c d+35 a^2 d^2\right )-d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x^2}{\sqrt {a+b x^2}} \, dx}{24 a b^3}\\ &=-\frac {d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt {a+b x^2}}{48 a b^4}-\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\left (d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{16 b^4}\\ &=-\frac {d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt {a+b x^2}}{48 a b^4}-\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\left (d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b^4}\\ &=-\frac {d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt {a+b x^2}}{48 a b^4}-\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 198, normalized size = 0.77 \begin {gather*} \frac {\frac {\sqrt {b} x \left (48 b^4 c^4+105 a^4 d^4+5 a^3 b d^3 \left (-72 c+7 d x^2\right )-2 a^2 b^2 d^2 \left (-216 c^2+60 c d x^2+7 d^2 x^4\right )+8 a b^3 d \left (-24 c^3+18 c^2 d x^2+6 c d^2 x^4+d^3 x^6\right )\right )}{a \sqrt {a+b x^2}}+3 d \left (-64 b^3 c^3+144 a b^2 c^2 d-120 a^2 b c d^2+35 a^3 d^3\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{48 b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[b]*x*(48*b^4*c^4 + 105*a^4*d^4 + 5*a^3*b*d^3*(-72*c + 7*d*x^2) - 2*a^2*b^2*d^2*(-216*c^2 + 60*c*d*x^2 +
7*d^2*x^4) + 8*a*b^3*d*(-24*c^3 + 18*c^2*d*x^2 + 6*c*d^2*x^4 + d^3*x^6)))/(a*Sqrt[a + b*x^2]) + 3*d*(-64*b^3*
c^3 + 144*a*b^2*c^2*d - 120*a^2*b*c*d^2 + 35*a^3*d^3)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(48*b^(9/2))

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Maple [A]
time = 0.10, size = 331, normalized size = 1.29

method	result	size
risch	\(\frac {d^{2} x \left (8 b^{2} d^{2} x^{4}-22 a b \,d^{2} x^{2}+48 b^{2} c d \,x^{2}+57 a^{2} d^{2}-168 a b c d +144 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}}{48 b^{4}}+\frac {x \,a^{3} d^{4}}{b^{4} \sqrt {b \,x^{2}+a}}-\frac {4 x \,a^{2} c \,d^{3}}{b^{3} \sqrt {b \,x^{2}+a}}+\frac {6 x a \,c^{2} d^{2}}{b^{2} \sqrt {b \,x^{2}+a}}-\frac {4 x \,c^{3} d}{b \sqrt {b \,x^{2}+a}}-\frac {35 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a^{3} d^{4}}{16 b^{\frac {9}{2}}}+\frac {15 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a^{2} c \,d^{3}}{2 b^{\frac {7}{2}}}-\frac {9 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a \,c^{2} d^{2}}{b^{\frac {5}{2}}}+\frac {4 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c^{3} d}{b^{\frac {3}{2}}}+\frac {c^{4} x}{a \sqrt {b \,x^{2}+a}}\)	\(282\)
default	\(d^{4} \left (\frac {x^{7}}{6 b \sqrt {b \,x^{2}+a}}-\frac {7 a \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )}{6 b}\right )+4 c \,d^{3} \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )+6 c^{2} d^{2} \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+4 c^{3} d \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\frac {c^{4} x}{a \sqrt {b \,x^{2}+a}}\)	\(331\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^4*(1/6*x^7/b/(b*x^2+a)^(1/2)-7/6*a/b*(1/4*x^5/b/(b*x^2+a)^(1/2)-5/4*a/b*(1/2*x^3/b/(b*x^2+a)^(1/2)-3/2*a/b*(
-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))))+4*c*d^3*(1/4*x^5/b/(b*x^2+a)^(1/2)-5/4*a/b*(1
/2*x^3/b/(b*x^2+a)^(1/2)-3/2*a/b*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2)))))+6*c^2*d^2*(1
/2*x^3/b/(b*x^2+a)^(1/2)-3/2*a/b*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))+4*c^3*d*(-x/b
/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2)))+c^4*x/a/(b*x^2+a)^(1/2)

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Maxima [A]
time = 0.27, size = 311, normalized size = 1.21 \begin {gather*} \frac {d^{4} x^{7}}{6 \, \sqrt {b x^{2} + a} b} + \frac {c d^{3} x^{5}}{\sqrt {b x^{2} + a} b} - \frac {7 \, a d^{4} x^{5}}{24 \, \sqrt {b x^{2} + a} b^{2}} + \frac {3 \, c^{2} d^{2} x^{3}}{\sqrt {b x^{2} + a} b} - \frac {5 \, a c d^{3} x^{3}}{2 \, \sqrt {b x^{2} + a} b^{2}} + \frac {35 \, a^{2} d^{4} x^{3}}{48 \, \sqrt {b x^{2} + a} b^{3}} + \frac {c^{4} x}{\sqrt {b x^{2} + a} a} - \frac {4 \, c^{3} d x}{\sqrt {b x^{2} + a} b} + \frac {9 \, a c^{2} d^{2} x}{\sqrt {b x^{2} + a} b^{2}} - \frac {15 \, a^{2} c d^{3} x}{2 \, \sqrt {b x^{2} + a} b^{3}} + \frac {35 \, a^{3} d^{4} x}{16 \, \sqrt {b x^{2} + a} b^{4}} + \frac {4 \, c^{3} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {9 \, a c^{2} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} + \frac {15 \, a^{2} c d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {7}{2}}} - \frac {35 \, a^{3} d^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*d^4*x^7/(sqrt(b*x^2 + a)*b) + c*d^3*x^5/(sqrt(b*x^2 + a)*b) - 7/24*a*d^4*x^5/(sqrt(b*x^2 + a)*b^2) + 3*c^2
*d^2*x^3/(sqrt(b*x^2 + a)*b) - 5/2*a*c*d^3*x^3/(sqrt(b*x^2 + a)*b^2) + 35/48*a^2*d^4*x^3/(sqrt(b*x^2 + a)*b^3)
+ c^4*x/(sqrt(b*x^2 + a)*a) - 4*c^3*d*x/(sqrt(b*x^2 + a)*b) + 9*a*c^2*d^2*x/(sqrt(b*x^2 + a)*b^2) - 15/2*a^2*
c*d^3*x/(sqrt(b*x^2 + a)*b^3) + 35/16*a^3*d^4*x/(sqrt(b*x^2 + a)*b^4) + 4*c^3*d*arcsinh(b*x/sqrt(a*b))/b^(3/2)
- 9*a*c^2*d^2*arcsinh(b*x/sqrt(a*b))/b^(5/2) + 15/2*a^2*c*d^3*arcsinh(b*x/sqrt(a*b))/b^(7/2) - 35/16*a^3*d^4*
arcsinh(b*x/sqrt(a*b))/b^(9/2)

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Fricas [A]
time = 0.58, size = 584, normalized size = 2.27 \begin {gather*} \left [-\frac {3 \, {\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} + {\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8 \, a b^{4} d^{4} x^{7} + 2 \, {\left (24 \, a b^{4} c d^{3} - 7 \, a^{2} b^{3} d^{4}\right )} x^{5} + {\left (144 \, a b^{4} c^{2} d^{2} - 120 \, a^{2} b^{3} c d^{3} + 35 \, a^{3} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (16 \, b^{5} c^{4} - 64 \, a b^{4} c^{3} d + 144 \, a^{2} b^{3} c^{2} d^{2} - 120 \, a^{3} b^{2} c d^{3} + 35 \, a^{4} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, {\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}, -\frac {3 \, {\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} + {\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, a b^{4} d^{4} x^{7} + 2 \, {\left (24 \, a b^{4} c d^{3} - 7 \, a^{2} b^{3} d^{4}\right )} x^{5} + {\left (144 \, a b^{4} c^{2} d^{2} - 120 \, a^{2} b^{3} c d^{3} + 35 \, a^{3} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (16 \, b^{5} c^{4} - 64 \, a b^{4} c^{3} d + 144 \, a^{2} b^{3} c^{2} d^{2} - 120 \, a^{3} b^{2} c d^{3} + 35 \, a^{4} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(3*(64*a^2*b^3*c^3*d - 144*a^3*b^2*c^2*d^2 + 120*a^4*b*c*d^3 - 35*a^5*d^4 + (64*a*b^4*c^3*d - 144*a^2*b
^3*c^2*d^2 + 120*a^3*b^2*c*d^3 - 35*a^4*b*d^4)*x^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) -
2*(8*a*b^4*d^4*x^7 + 2*(24*a*b^4*c*d^3 - 7*a^2*b^3*d^4)*x^5 + (144*a*b^4*c^2*d^2 - 120*a^2*b^3*c*d^3 + 35*a^3*
b^2*d^4)*x^3 + 3*(16*b^5*c^4 - 64*a*b^4*c^3*d + 144*a^2*b^3*c^2*d^2 - 120*a^3*b^2*c*d^3 + 35*a^4*b*d^4)*x)*sqr
t(b*x^2 + a))/(a*b^6*x^2 + a^2*b^5), -1/48*(3*(64*a^2*b^3*c^3*d - 144*a^3*b^2*c^2*d^2 + 120*a^4*b*c*d^3 - 35*a
^5*d^4 + (64*a*b^4*c^3*d - 144*a^2*b^3*c^2*d^2 + 120*a^3*b^2*c*d^3 - 35*a^4*b*d^4)*x^2)*sqrt(-b)*arctan(sqrt(-
b)*x/sqrt(b*x^2 + a)) - (8*a*b^4*d^4*x^7 + 2*(24*a*b^4*c*d^3 - 7*a^2*b^3*d^4)*x^5 + (144*a*b^4*c^2*d^2 - 120*a
^2*b^3*c*d^3 + 35*a^3*b^2*d^4)*x^3 + 3*(16*b^5*c^4 - 64*a*b^4*c^3*d + 144*a^2*b^3*c^2*d^2 - 120*a^3*b^2*c*d^3
+ 35*a^4*b*d^4)*x)*sqrt(b*x^2 + a))/(a*b^6*x^2 + a^2*b^5)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{4}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.65, size = 235, normalized size = 0.91 \begin {gather*} \frac {{\left ({\left (2 \, {\left (\frac {4 \, d^{4} x^{2}}{b} + \frac {24 \, a b^{6} c d^{3} - 7 \, a^{2} b^{5} d^{4}}{a b^{7}}\right )} x^{2} + \frac {144 \, a b^{6} c^{2} d^{2} - 120 \, a^{2} b^{5} c d^{3} + 35 \, a^{3} b^{4} d^{4}}{a b^{7}}\right )} x^{2} + \frac {3 \, {\left (16 \, b^{7} c^{4} - 64 \, a b^{6} c^{3} d + 144 \, a^{2} b^{5} c^{2} d^{2} - 120 \, a^{3} b^{4} c d^{3} + 35 \, a^{4} b^{3} d^{4}\right )}}{a b^{7}}\right )} x}{48 \, \sqrt {b x^{2} + a}} - \frac {{\left (64 \, b^{3} c^{3} d - 144 \, a b^{2} c^{2} d^{2} + 120 \, a^{2} b c d^{3} - 35 \, a^{3} d^{4}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*((2*(4*d^4*x^2/b + (24*a*b^6*c*d^3 - 7*a^2*b^5*d^4)/(a*b^7))*x^2 + (144*a*b^6*c^2*d^2 - 120*a^2*b^5*c*d^3
+ 35*a^3*b^4*d^4)/(a*b^7))*x^2 + 3*(16*b^7*c^4 - 64*a*b^6*c^3*d + 144*a^2*b^5*c^2*d^2 - 120*a^3*b^4*c*d^3 + 3
5*a^4*b^3*d^4)/(a*b^7))*x/sqrt(b*x^2 + a) - 1/16*(64*b^3*c^3*d - 144*a*b^2*c^2*d^2 + 120*a^2*b*c*d^3 - 35*a^3*
d^4)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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