3.5.15 \(\int \frac {(a+c x^2)^4}{d+e x} \, dx\)

Optimal. Leaf size=264 \[ \frac {c^2 (d+e x)^4 \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{2 e^9}+\frac {2 c^3 (d+e x)^6 \left (a e^2+7 c d^2\right )}{3 e^9}-\frac {8 c^3 d (d+e x)^5 \left (3 a e^2+7 c d^2\right )}{5 e^9}-\frac {8 c^2 d (d+e x)^3 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{3 e^9}+\frac {2 c (d+e x)^2 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{e^9}+\frac {\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9}-\frac {8 c d x \left (a e^2+c d^2\right )^3}{e^8}+\frac {c^4 (d+e x)^8}{8 e^9}-\frac {8 c^4 d (d+e x)^7}{7 e^9} \]

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Rubi [A]  time = 0.28, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \begin {gather*} \frac {c^2 (d+e x)^4 \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{2 e^9}+\frac {2 c^3 (d+e x)^6 \left (a e^2+7 c d^2\right )}{3 e^9}-\frac {8 c^3 d (d+e x)^5 \left (3 a e^2+7 c d^2\right )}{5 e^9}-\frac {8 c^2 d (d+e x)^3 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{3 e^9}+\frac {2 c (d+e x)^2 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{e^9}-\frac {8 c d x \left (a e^2+c d^2\right )^3}{e^8}+\frac {\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9}+\frac {c^4 (d+e x)^8}{8 e^9}-\frac {8 c^4 d (d+e x)^7}{7 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^4/(d + e*x),x]

[Out]

(-8*c*d*(c*d^2 + a*e^2)^3*x)/e^8 + (2*c*(c*d^2 + a*e^2)^2*(7*c*d^2 + a*e^2)*(d + e*x)^2)/e^9 - (8*c^2*d*(c*d^2
+ a*e^2)*(7*c*d^2 + 3*a*e^2)*(d + e*x)^3)/(3*e^9) + (c^2*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)*(d + e*x)^
4)/(2*e^9) - (8*c^3*d*(7*c*d^2 + 3*a*e^2)*(d + e*x)^5)/(5*e^9) + (2*c^3*(7*c*d^2 + a*e^2)*(d + e*x)^6)/(3*e^9)
- (8*c^4*d*(d + e*x)^7)/(7*e^9) + (c^4*(d + e*x)^8)/(8*e^9) + ((c*d^2 + a*e^2)^4*Log[d + e*x])/e^9

Rule 697

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

Rubi steps

\begin {align*} \int \frac {\left (a+c x^2\right )^4}{d+e x} \, dx &=\int \left (-\frac {8 c d \left (c d^2+a e^2\right )^3}{e^8}+\frac {\left (c d^2+a e^2\right )^4}{e^8 (d+e x)}+\frac {4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)}{e^8}+\frac {8 c^2 d \left (-7 c d^2-3 a e^2\right ) \left (c d^2+a e^2\right ) (d+e x)^2}{e^8}+\frac {2 c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^3}{e^8}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^4}{e^8}+\frac {4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^5}{e^8}-\frac {8 c^4 d (d+e x)^6}{e^8}+\frac {c^4 (d+e x)^7}{e^8}\right ) \, dx\\ &=-\frac {8 c d \left (c d^2+a e^2\right )^3 x}{e^8}+\frac {2 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^2}{e^9}-\frac {8 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^3}{3 e^9}+\frac {c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^4}{2 e^9}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^5}{5 e^9}+\frac {2 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^6}{3 e^9}-\frac {8 c^4 d (d+e x)^7}{7 e^9}+\frac {c^4 (d+e x)^8}{8 e^9}+\frac {\left (c d^2+a e^2\right )^4 \log (d+e x)}{e^9}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 227, normalized size = 0.86 \begin {gather*} \frac {c x \left (1680 a^3 e^6 (e x-2 d)+420 a^2 c e^4 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+56 a c^2 e^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+c^3 \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )\right )}{840 e^8}+\frac {\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^4/(d + e*x),x]

[Out]

(c*x*(1680*a^3*e^6*(-2*d + e*x) + 420*a^2*c*e^4*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 56*a*c^2*e^2
*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + c^3*(-840*d^7 + 420*d^
6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7)))
/(840*e^8) + ((c*d^2 + a*e^2)^4*Log[d + e*x])/e^9

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + c*x^2)^4/(d + e*x),x]

[Out]

IntegrateAlgebraic[(a + c*x^2)^4/(d + e*x), x]

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fricas [A]  time = 0.38, size = 319, normalized size = 1.21 \begin {gather*} \frac {105 \, c^{4} e^{8} x^{8} - 120 \, c^{4} d e^{7} x^{7} + 140 \, {\left (c^{4} d^{2} e^{6} + 4 \, a c^{3} e^{8}\right )} x^{6} - 168 \, {\left (c^{4} d^{3} e^{5} + 4 \, a c^{3} d e^{7}\right )} x^{5} + 210 \, {\left (c^{4} d^{4} e^{4} + 4 \, a c^{3} d^{2} e^{6} + 6 \, a^{2} c^{2} e^{8}\right )} x^{4} - 280 \, {\left (c^{4} d^{5} e^{3} + 4 \, a c^{3} d^{3} e^{5} + 6 \, a^{2} c^{2} d e^{7}\right )} x^{3} + 420 \, {\left (c^{4} d^{6} e^{2} + 4 \, a c^{3} d^{4} e^{4} + 6 \, a^{2} c^{2} d^{2} e^{6} + 4 \, a^{3} c e^{8}\right )} x^{2} - 840 \, {\left (c^{4} d^{7} e + 4 \, a c^{3} d^{5} e^{3} + 6 \, a^{2} c^{2} d^{3} e^{5} + 4 \, a^{3} c d e^{7}\right )} x + 840 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (e x + d\right )}{840 \, e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(105*c^4*e^8*x^8 - 120*c^4*d*e^7*x^7 + 140*(c^4*d^2*e^6 + 4*a*c^3*e^8)*x^6 - 168*(c^4*d^3*e^5 + 4*a*c^3*
d*e^7)*x^5 + 210*(c^4*d^4*e^4 + 4*a*c^3*d^2*e^6 + 6*a^2*c^2*e^8)*x^4 - 280*(c^4*d^5*e^3 + 4*a*c^3*d^3*e^5 + 6*
a^2*c^2*d*e^7)*x^3 + 420*(c^4*d^6*e^2 + 4*a*c^3*d^4*e^4 + 6*a^2*c^2*d^2*e^6 + 4*a^3*c*e^8)*x^2 - 840*(c^4*d^7*
e + 4*a*c^3*d^5*e^3 + 6*a^2*c^2*d^3*e^5 + 4*a^3*c*d*e^7)*x + 840*(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^
4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*log(e*x + d))/e^9

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giac [A]  time = 0.16, size = 316, normalized size = 1.20 \begin {gather*} {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{840} \, {\left (105 \, c^{4} x^{8} e^{7} - 120 \, c^{4} d x^{7} e^{6} + 140 \, c^{4} d^{2} x^{6} e^{5} - 168 \, c^{4} d^{3} x^{5} e^{4} + 210 \, c^{4} d^{4} x^{4} e^{3} - 280 \, c^{4} d^{5} x^{3} e^{2} + 420 \, c^{4} d^{6} x^{2} e - 840 \, c^{4} d^{7} x + 560 \, a c^{3} x^{6} e^{7} - 672 \, a c^{3} d x^{5} e^{6} + 840 \, a c^{3} d^{2} x^{4} e^{5} - 1120 \, a c^{3} d^{3} x^{3} e^{4} + 1680 \, a c^{3} d^{4} x^{2} e^{3} - 3360 \, a c^{3} d^{5} x e^{2} + 1260 \, a^{2} c^{2} x^{4} e^{7} - 1680 \, a^{2} c^{2} d x^{3} e^{6} + 2520 \, a^{2} c^{2} d^{2} x^{2} e^{5} - 5040 \, a^{2} c^{2} d^{3} x e^{4} + 1680 \, a^{3} c x^{2} e^{7} - 3360 \, a^{3} c d x e^{6}\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*e^(-9)*log(abs(x*e + d)) + 1/840*(
105*c^4*x^8*e^7 - 120*c^4*d*x^7*e^6 + 140*c^4*d^2*x^6*e^5 - 168*c^4*d^3*x^5*e^4 + 210*c^4*d^4*x^4*e^3 - 280*c^
4*d^5*x^3*e^2 + 420*c^4*d^6*x^2*e - 840*c^4*d^7*x + 560*a*c^3*x^6*e^7 - 672*a*c^3*d*x^5*e^6 + 840*a*c^3*d^2*x^
4*e^5 - 1120*a*c^3*d^3*x^3*e^4 + 1680*a*c^3*d^4*x^2*e^3 - 3360*a*c^3*d^5*x*e^2 + 1260*a^2*c^2*x^4*e^7 - 1680*a
^2*c^2*d*x^3*e^6 + 2520*a^2*c^2*d^2*x^2*e^5 - 5040*a^2*c^2*d^3*x*e^4 + 1680*a^3*c*x^2*e^7 - 3360*a^3*c*d*x*e^6
)*e^(-8)

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maple [A]  time = 0.04, size = 358, normalized size = 1.36 \begin {gather*} \frac {c^{4} x^{8}}{8 e}-\frac {c^{4} d \,x^{7}}{7 e^{2}}+\frac {2 a \,c^{3} x^{6}}{3 e}+\frac {c^{4} d^{2} x^{6}}{6 e^{3}}-\frac {4 a \,c^{3} d \,x^{5}}{5 e^{2}}-\frac {c^{4} d^{3} x^{5}}{5 e^{4}}+\frac {3 a^{2} c^{2} x^{4}}{2 e}+\frac {a \,c^{3} d^{2} x^{4}}{e^{3}}+\frac {c^{4} d^{4} x^{4}}{4 e^{5}}-\frac {2 a^{2} c^{2} d \,x^{3}}{e^{2}}-\frac {4 a \,c^{3} d^{3} x^{3}}{3 e^{4}}-\frac {c^{4} d^{5} x^{3}}{3 e^{6}}+\frac {2 a^{3} c \,x^{2}}{e}+\frac {3 a^{2} c^{2} d^{2} x^{2}}{e^{3}}+\frac {2 a \,c^{3} d^{4} x^{2}}{e^{5}}+\frac {c^{4} d^{6} x^{2}}{2 e^{7}}+\frac {a^{4} \ln \left (e x +d \right )}{e}+\frac {4 a^{3} c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {4 a^{3} c d x}{e^{2}}+\frac {6 a^{2} c^{2} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {6 a^{2} c^{2} d^{3} x}{e^{4}}+\frac {4 a \,c^{3} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {4 a \,c^{3} d^{5} x}{e^{6}}+\frac {c^{4} d^{8} \ln \left (e x +d \right )}{e^{9}}-\frac {c^{4} d^{7} x}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/5*c^3/e^2*x^5*a*d+c^3/e^3*x^4*a*d^2+1/6*c^4/e^3*x^6*d^2-1/5*c^4/e^4*x^5*d^3+3/2*c^2/e*x^4*a^2-1/7*c^4/e^2*d
*x^7-c^4/e^8*d^7*x+2/3*c^3/e*x^6*a+1/4*c^4/e^5*x^4*d^4-1/3*c^4/e^6*x^3*d^5+2*c/e*x^2*a^3+1/2*c^4/e^7*x^2*d^6+1
/e^9*ln(e*x+d)*c^4*d^8+1/e*ln(e*x+d)*a^4+4/e^3*ln(e*x+d)*a^3*c*d^2+6/e^5*ln(e*x+d)*a^2*c^2*d^4+4/e^7*ln(e*x+d)
*a*c^3*d^6-2*c^2/e^2*x^3*a^2*d-4*c^3/e^6*a*d^5*x+1/8*c^4/e*x^8-6*c^2/e^4*a^2*d^3*x+2*c^3/e^5*x^2*a*d^4-4*c/e^2
*a^3*d*x-4/3*c^3/e^4*x^3*a*d^3+3*c^2/e^3*x^2*a^2*d^2

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maxima [A]  time = 1.33, size = 319, normalized size = 1.21 \begin {gather*} \frac {105 \, c^{4} e^{7} x^{8} - 120 \, c^{4} d e^{6} x^{7} + 140 \, {\left (c^{4} d^{2} e^{5} + 4 \, a c^{3} e^{7}\right )} x^{6} - 168 \, {\left (c^{4} d^{3} e^{4} + 4 \, a c^{3} d e^{6}\right )} x^{5} + 210 \, {\left (c^{4} d^{4} e^{3} + 4 \, a c^{3} d^{2} e^{5} + 6 \, a^{2} c^{2} e^{7}\right )} x^{4} - 280 \, {\left (c^{4} d^{5} e^{2} + 4 \, a c^{3} d^{3} e^{4} + 6 \, a^{2} c^{2} d e^{6}\right )} x^{3} + 420 \, {\left (c^{4} d^{6} e + 4 \, a c^{3} d^{4} e^{3} + 6 \, a^{2} c^{2} d^{2} e^{5} + 4 \, a^{3} c e^{7}\right )} x^{2} - 840 \, {\left (c^{4} d^{7} + 4 \, a c^{3} d^{5} e^{2} + 6 \, a^{2} c^{2} d^{3} e^{4} + 4 \, a^{3} c d e^{6}\right )} x}{840 \, e^{8}} + \frac {{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (e x + d\right )}{e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(105*c^4*e^7*x^8 - 120*c^4*d*e^6*x^7 + 140*(c^4*d^2*e^5 + 4*a*c^3*e^7)*x^6 - 168*(c^4*d^3*e^4 + 4*a*c^3*
d*e^6)*x^5 + 210*(c^4*d^4*e^3 + 4*a*c^3*d^2*e^5 + 6*a^2*c^2*e^7)*x^4 - 280*(c^4*d^5*e^2 + 4*a*c^3*d^3*e^4 + 6*
a^2*c^2*d*e^6)*x^3 + 420*(c^4*d^6*e + 4*a*c^3*d^4*e^3 + 6*a^2*c^2*d^2*e^5 + 4*a^3*c*e^7)*x^2 - 840*(c^4*d^7 +
4*a*c^3*d^5*e^2 + 6*a^2*c^2*d^3*e^4 + 4*a^3*c*d*e^6)*x)/e^8 + (c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 +
4*a^3*c*d^2*e^6 + a^4*e^8)*log(e*x + d)/e^9

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mupad [B]  time = 0.07, size = 357, normalized size = 1.35 \begin {gather*} x^2\,\left (\frac {d^2\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{e^2}+\frac {6\,a^2\,c^2}{e}\right )}{2\,e^2}+\frac {2\,a^3\,c}{e}\right )+x^4\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{4\,e^2}+\frac {3\,a^2\,c^2}{2\,e}\right )+x^6\,\left (\frac {2\,a\,c^3}{3\,e}+\frac {c^4\,d^2}{6\,e^3}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^4\,e^8+4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4+4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{e^9}+\frac {c^4\,x^8}{8\,e}-\frac {c^4\,d\,x^7}{7\,e^2}-\frac {d\,x^3\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{e^2}+\frac {6\,a^2\,c^2}{e}\right )}{3\,e}-\frac {d\,x^5\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{5\,e}-\frac {d\,x\,\left (\frac {d^2\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{e^2}+\frac {6\,a^2\,c^2}{e}\right )}{e^2}+\frac {4\,a^3\,c}{e}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^4/(d + e*x),x)

[Out]

x^2*((d^2*((d^2*((4*a*c^3)/e + (c^4*d^2)/e^3))/e^2 + (6*a^2*c^2)/e))/(2*e^2) + (2*a^3*c)/e) + x^4*((d^2*((4*a*
c^3)/e + (c^4*d^2)/e^3))/(4*e^2) + (3*a^2*c^2)/(2*e)) + x^6*((2*a*c^3)/(3*e) + (c^4*d^2)/(6*e^3)) + (log(d + e
*x)*(a^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4))/e^9 + (c^4*x^8)/(8*e) - (c^4*
d*x^7)/(7*e^2) - (d*x^3*((d^2*((4*a*c^3)/e + (c^4*d^2)/e^3))/e^2 + (6*a^2*c^2)/e))/(3*e) - (d*x^5*((4*a*c^3)/e
+ (c^4*d^2)/e^3))/(5*e) - (d*x*((d^2*((d^2*((4*a*c^3)/e + (c^4*d^2)/e^3))/e^2 + (6*a^2*c^2)/e))/e^2 + (4*a^3*
c)/e))/e

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sympy [A]  time = 0.57, size = 292, normalized size = 1.11 \begin {gather*} - \frac {c^{4} d x^{7}}{7 e^{2}} + \frac {c^{4} x^{8}}{8 e} + x^{6} \left (\frac {2 a c^{3}}{3 e} + \frac {c^{4} d^{2}}{6 e^{3}}\right ) + x^{5} \left (- \frac {4 a c^{3} d}{5 e^{2}} - \frac {c^{4} d^{3}}{5 e^{4}}\right ) + x^{4} \left (\frac {3 a^{2} c^{2}}{2 e} + \frac {a c^{3} d^{2}}{e^{3}} + \frac {c^{4} d^{4}}{4 e^{5}}\right ) + x^{3} \left (- \frac {2 a^{2} c^{2} d}{e^{2}} - \frac {4 a c^{3} d^{3}}{3 e^{4}} - \frac {c^{4} d^{5}}{3 e^{6}}\right ) + x^{2} \left (\frac {2 a^{3} c}{e} + \frac {3 a^{2} c^{2} d^{2}}{e^{3}} + \frac {2 a c^{3} d^{4}}{e^{5}} + \frac {c^{4} d^{6}}{2 e^{7}}\right ) + x \left (- \frac {4 a^{3} c d}{e^{2}} - \frac {6 a^{2} c^{2} d^{3}}{e^{4}} - \frac {4 a c^{3} d^{5}}{e^{6}} - \frac {c^{4} d^{7}}{e^{8}}\right ) + \frac {\left (a e^{2} + c d^{2}\right )^{4} \log {\left (d + e x \right )}}{e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c**4*d*x**7/(7*e**2) + c**4*x**8/(8*e) + x**6*(2*a*c**3/(3*e) + c**4*d**2/(6*e**3)) + x**5*(-4*a*c**3*d/(5*e*
*2) - c**4*d**3/(5*e**4)) + x**4*(3*a**2*c**2/(2*e) + a*c**3*d**2/e**3 + c**4*d**4/(4*e**5)) + x**3*(-2*a**2*c
**2*d/e**2 - 4*a*c**3*d**3/(3*e**4) - c**4*d**5/(3*e**6)) + x**2*(2*a**3*c/e + 3*a**2*c**2*d**2/e**3 + 2*a*c**
3*d**4/e**5 + c**4*d**6/(2*e**7)) + x*(-4*a**3*c*d/e**2 - 6*a**2*c**2*d**3/e**4 - 4*a*c**3*d**5/e**6 - c**4*d*
*7/e**8) + (a*e**2 + c*d**2)**4*log(d + e*x)/e**9

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