Optimal. Leaf size=267 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {p x^3+q} \left (\sqrt {2} a \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {2} b \sqrt [4]{c} \sqrt [4]{d}\right )}{a^2 \sqrt {c} x^2+2 a b \sqrt {c} x+b^2 \sqrt {c}+\sqrt {d} p x^3+\sqrt {d} q}\right )}{\sqrt {2} c^{3/4} \sqrt [4]{d}}+\frac {\tan ^{-1}\left (\frac {a \sqrt [4]{c} x+b \sqrt [4]{c}}{a \sqrt [4]{c} x+b \sqrt [4]{c}-\sqrt {2} \sqrt [4]{d} \sqrt {p x^3+q}}\right )}{\sqrt {2} c^{3/4} \sqrt [4]{d}}-\frac {\tan ^{-1}\left (\frac {a \sqrt [4]{c} x+b \sqrt [4]{c}}{a \sqrt [4]{c} x+b \sqrt [4]{c}+\sqrt {2} \sqrt [4]{d} \sqrt {p x^3+q}}\right )}{\sqrt {2} c^{3/4} \sqrt [4]{d}} \]
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Rubi [F] time = 13.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx &=\int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx\\ &=\int \left (\frac {2 a b^2 q}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )}+\frac {4 a^2 b q x}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )}+\frac {2 a^3 \left (1-\frac {3 b^3 p}{2 a^3 q}\right ) q x^2}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )}+\frac {7 a b^2 p x^3}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )}+\frac {5 a^2 b p x^4}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )}+\frac {a^3 p x^5}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )}\right ) \, dx\\ &=\left (a^3 p\right ) \int \frac {x^5}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx+\left (5 a^2 b p\right ) \int \frac {x^4}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx+\left (7 a b^2 p\right ) \int \frac {x^3}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx+\left (4 a^2 b q\right ) \int \frac {x}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )} \, dx+\left (2 a b^2 q\right ) \int \frac {1}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )} \, dx+\left (-3 b^3 p+2 a^3 q\right ) \int \frac {x^2}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )} \, dx\\ &=\left (a^3 p\right ) \int \frac {x^5}{\sqrt {q+p x^3} \left (b^4 c+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c x^3+a^4 c x^4+d \left (q+p x^3\right )^2\right )} \, dx+\left (5 a^2 b p\right ) \int \frac {x^4}{\sqrt {q+p x^3} \left (b^4 c+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c x^3+a^4 c x^4+d \left (q+p x^3\right )^2\right )} \, dx+\left (7 a b^2 p\right ) \int \frac {x^3}{\sqrt {q+p x^3} \left (b^4 c+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c x^3+a^4 c x^4+d \left (q+p x^3\right )^2\right )} \, dx+\left (4 a^2 b q\right ) \int \frac {x}{\sqrt {q+p x^3} \left (-b^4 c-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c x^3-a^4 c x^4-d \left (q+p x^3\right )^2\right )} \, dx+\left (2 a b^2 q\right ) \int \frac {1}{\sqrt {q+p x^3} \left (-b^4 c-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c x^3-a^4 c x^4-d \left (q+p x^3\right )^2\right )} \, dx+\left (-3 b^3 p+2 a^3 q\right ) \int \frac {x^2}{\sqrt {q+p x^3} \left (-b^4 c-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c x^3-a^4 c x^4-d \left (q+p x^3\right )^2\right )} \, dx\\ \end {align*}
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Mathematica [C] time = 7.40, size = 52549, normalized size = 196.81 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 21.74, size = 267, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {b \sqrt [4]{c}+a \sqrt [4]{c} x}{b \sqrt [4]{c}+a \sqrt [4]{c} x-\sqrt {2} \sqrt [4]{d} \sqrt {q+p x^3}}\right )}{\sqrt {2} c^{3/4} \sqrt [4]{d}}-\frac {\tan ^{-1}\left (\frac {b \sqrt [4]{c}+a \sqrt [4]{c} x}{b \sqrt [4]{c}+a \sqrt [4]{c} x+\sqrt {2} \sqrt [4]{d} \sqrt {q+p x^3}}\right )}{\sqrt {2} c^{3/4} \sqrt [4]{d}}+\frac {\tanh ^{-1}\left (\frac {\left (\sqrt {2} b \sqrt [4]{c} \sqrt [4]{d}+\sqrt {2} a \sqrt [4]{c} \sqrt [4]{d} x\right ) \sqrt {q+p x^3}}{b^2 \sqrt {c}+\sqrt {d} q+2 a b \sqrt {c} x+a^2 \sqrt {c} x^2+\sqrt {d} p x^3}\right )}{\sqrt {2} c^{3/4} \sqrt [4]{d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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.
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giac [F(-1)] 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.
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maple [C] time = 3.00, size = 7274, normalized size = 27.24
method | result | size |
default | \(\text {Expression too large to display}\) | \(7274\) |
elliptic | \(\text {Expression too large to display}\) | \(7274\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a p x^{3} + 3 \, b p x^{2} - 2 \, a q\right )} {\left (a x + b\right )}^{2}}{{\left (a^{4} c x^{4} + d p^{2} x^{6} + 6 \, a^{2} b^{2} c x^{2} + 4 \, a b^{3} c x + b^{4} c + 2 \, {\left (2 \, a^{3} b c + d p q\right )} x^{3} + d q^{2}\right )} \sqrt {p x^{3} + q}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x + b\right )^{2} \left (a p x^{3} - 2 a q + 3 b p x^{2}\right )}{\sqrt {p x^{3} + q} \left (a^{4} c x^{4} + 4 a^{3} b c x^{3} + 6 a^{2} b^{2} c x^{2} + 4 a b^{3} c x + b^{4} c + d p^{2} x^{6} + 2 d p q x^{3} + d q^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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