Optimal. Leaf size=42 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {p x^3+q}}{\sqrt {c} (a x+b)}\right )}{\sqrt {c} \sqrt {d}} \]
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Rubi [F] time = 2.63, 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 {-2 a q+3 b p x^2+a p x^3}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {-2 a q+3 b p x^2+a p x^3}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )} \, dx &=\int \left (\frac {a}{d \sqrt {q+p x^3}}-\frac {a \left (b^2 c+3 d q\right )+2 a^2 b c x+\left (a^3 c-3 b d p\right ) x^2}{d \sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )}\right ) \, dx\\ &=-\frac {\int \frac {a \left (b^2 c+3 d q\right )+2 a^2 b c x+\left (a^3 c-3 b d p\right ) x^2}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )} \, dx}{d}+\frac {a \int \frac {1}{\sqrt {q+p x^3}} \, dx}{d}\\ &=\frac {2 \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{q}+\sqrt [3]{p} x\right ) \sqrt {\frac {q^{2/3}-\sqrt [3]{p} \sqrt [3]{q} x+p^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d \sqrt [3]{p} \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x\right )^2}} \sqrt {q+p x^3}}-\frac {\int \left (\frac {a \left (b^2 c+3 d q\right )}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )}+\frac {2 a^2 b c x}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )}+\frac {\left (a^3 c-3 b d p\right ) x^2}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )}\right ) \, dx}{d}\\ &=\frac {2 \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{q}+\sqrt [3]{p} x\right ) \sqrt {\frac {q^{2/3}-\sqrt [3]{p} \sqrt [3]{q} x+p^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d \sqrt [3]{p} \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x\right )^2}} \sqrt {q+p x^3}}-\frac {\left (2 a^2 b c\right ) \int \frac {x}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )} \, dx}{d}-\frac {\left (a^3 c-3 b d p\right ) \int \frac {x^2}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )} \, dx}{d}-\frac {\left (a \left (b^2 c+3 d q\right )\right ) \int \frac {1}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right )} \, dx}{d}\\ \end {align*}
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Mathematica [C] time = 6.79, size = 5105, normalized size = 121.55 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 6.70, size = 42, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {q+p x^3}}{\sqrt {c} (b+a x)}\right )}{\sqrt {c} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.68, size = 459, normalized size = 10.93 \begin {gather*} \left [-\frac {\sqrt {-c d} \log \left (-\frac {6 \, a^{2} c d p x^{5} - d^{2} p^{2} x^{6} - b^{4} c^{2} + 6 \, b^{2} c d q - {\left (a^{4} c^{2} - 12 \, a b c d p\right )} x^{4} - d^{2} q^{2} - 2 \, {\left (2 \, a^{3} b c^{2} - 3 \, b^{2} c d p + d^{2} p q\right )} x^{3} - 6 \, {\left (a^{2} b^{2} c^{2} - a^{2} c d q\right )} x^{2} + 4 \, {\left (a d p x^{4} - 3 \, a^{2} b c x^{2} - b^{3} c - {\left (a^{3} c - b d p\right )} x^{3} + b d q - {\left (3 \, a b^{2} c - a d q\right )} x\right )} \sqrt {p x^{3} + q} \sqrt {-c d} - 4 \, {\left (a b^{3} c^{2} - 3 \, a b c d q\right )} x}{2 \, a^{2} c d p x^{5} + d^{2} p^{2} x^{6} + b^{4} c^{2} + 2 \, b^{2} c d q + {\left (a^{4} c^{2} + 4 \, a b c d p\right )} x^{4} + d^{2} q^{2} + 2 \, {\left (2 \, a^{3} b c^{2} + b^{2} c d p + d^{2} p q\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} c^{2} + a^{2} c d q\right )} x^{2} + 4 \, {\left (a b^{3} c^{2} + a b c d q\right )} x}\right )}{2 \, c d}, \frac {\sqrt {c d} \arctan \left (-\frac {{\left (a^{2} c x^{2} - d p x^{3} + 2 \, a b c x + b^{2} c - d q\right )} \sqrt {p x^{3} + q} \sqrt {c d}}{2 \, {\left (a c d p x^{4} + b c d p x^{3} + a c d q x + b c d q\right )}}\right )}{c d}\right ] \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 = 0.77, size = 2262, normalized size = 53.86
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(2262\) |
default | \(\text {Expression too large to display}\) | \(2264\) |
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 {a p x^{3} + 3 \, b p x^{2} - 2 \, a q}{{\left (a^{2} c x^{2} + d p x^{3} + 2 \, a b c x + b^{2} c + d q\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 [B] time = 10.90, size = 456, normalized size = 10.86 \begin {gather*} \frac {\ln \left (\frac {\left (a^3\,b\,c\,1{}\mathrm {i}-b^2\,d\,p\,1{}\mathrm {i}+a^4\,c\,x\,1{}\mathrm {i}+2\,a^3\,\sqrt {c}\,\sqrt {d}\,\sqrt {p\,x^3+q}-a^2\,d\,p\,x^2\,1{}\mathrm {i}+a\,b\,d\,p\,x\,1{}\mathrm {i}\right )\,\left (a^3\,b^3\,c^2\,1{}\mathrm {i}+a^6\,c^2\,x^3\,1{}\mathrm {i}+a^4\,b^2\,c^2\,x\,3{}\mathrm {i}+a^5\,b\,c^2\,x^2\,3{}\mathrm {i}-b^4\,c\,d\,p\,1{}\mathrm {i}+b^2\,d^2\,p\,\left (p\,x^3+q\right )\,1{}\mathrm {i}+a^2\,d^2\,p\,x^2\,\left (p\,x^3+q\right )\,1{}\mathrm {i}+a^3\,b\,c\,d\,q\,1{}\mathrm {i}+a^3\,b\,c\,d\,\left (p\,x^3+q\right )\,2{}\mathrm {i}+a^4\,c\,d\,q\,x\,1{}\mathrm {i}+a^4\,c\,d\,x\,\left (p\,x^3+q\right )\,2{}\mathrm {i}+2\,a^3\,\sqrt {c}\,d^{3/2}\,q\,\sqrt {p\,x^3+q}-2\,b^3\,\sqrt {c}\,d^{3/2}\,p\,\sqrt {p\,x^3+q}-a\,b^3\,c\,d\,p\,x\,1{}\mathrm {i}-a\,b\,d^2\,p\,x\,\left (p\,x^3+q\right )\,1{}\mathrm {i}\right )}{\left (c\,a^2\,x^2+2\,c\,a\,b\,x+c\,b^2+d\,p\,x^3+d\,q\right )\,\left (a^8\,c^2\,x^2+2\,a^7\,b\,c^2\,x+a^6\,b^2\,c^2+2\,a^6\,c\,d\,p\,x^3+4\,q\,a^6\,c\,d+a^4\,d^2\,p^2\,x^4-2\,a^3\,b^3\,c\,d\,p-2\,a^3\,b\,d^2\,p^2\,x^3+3\,a^2\,b^2\,d^2\,p^2\,x^2-2\,a\,b^3\,d^2\,p^2\,x+b^4\,d^2\,p^2\right )}\right )\,1{}\mathrm {i}}{\sqrt {c}\,\sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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